Function Spaces

CONTEMPORARY MATHEMATICS 328

Function Spaces Fourth Conference on Function Spaces May 14-19,2002 Southern Illinois University at Edwardsville Krzysztof Jarosz Editor

Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

http://dx.doi.org/10.1090/conm/328

CoNTEMPORARY MATHEMATICS 328

Function Spaces Fourth Conference on Function Spaces May 14-19, 2002 Southern Illinois University at Edwardsville

Krzysztof Jarosz Editor

Licensed to AMS.

American Mathematical Society Providence, Rhode Island

License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Editorial Board Dennis DeTurck, managing editor Andreas Blass

Andy R. Magid

Michael Vogelius

This volume contains the proceedings of the Fourth Conference on Function Spaces, held May 14~ 19, 2002, at Southern Illinois University at Edwardsville. 2000 Mathematics Subject Classification. Primary 32H02, 46E25, 46H05, 46J10, 46J15, 46L07, 47 AlO, 47B38, 47L10, 54D05.

Library of Congress Cataloging-in-Publication Data Conference on Function Space (4th : 2002 : Southern Illinois University at Edwardsville) Function spaces : Fourth Conference on Function Spaces, May 14-19, 2002, Southern Illinois University at Edwardsville / Krzysztof Jarosz, editor. p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 328) Includes bibliographical references. ISBN 0-8218-3269-7 (softcover: alk. paper) 1. Function spaces-Congresses. I. Jarosz, Krzysztof, 1953~ II. Title. III. Contemporary mathematics (American Mathematical Society) ; v. 328 QA323.C66 2002 515'.73-dc21

2003045306

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Contents

Preface

v

Components of resolvent sets and local spectral theory PIETRO AlENA AND FERNANDO VILLAFANE

The Fejer-Riesz inequality and the index of the shift JOHN R. AKEROYD

A Cauchy-Green formula on the unit sphere in C 2 JOHN T. ANDERSON AND JOHN WERMER

On a-dual algebras HUGO ARIZMENDI,

ANGEL CARRILLO, AND LOURDES PALACIOS

1

15 21

31

A connected metric space that is not separably connected RICHARD M. ARON AND MANUEL MAESTRE

Weighted Chebyshev centres and intersection properties of balls in Banach spaces PRADIPTA BANDYOPADHYAY AND S. DUTTA The boundary of the unit ball in H 1 -type spaces PAUL BENEKER AND JAN WIEGERINCK

Complete isometries - an illustration of noncommutative functional analysis DAVID P. BLECHER AND DAMON M. HAY

39

43 59

85

Some recent trends and advances in certain lattice ordered algebras KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces EGGERT BRIEM

Some mapping properties of p-summing operators with Hilbertian domain QrNGYING Bu The unique decomposition property and the Banach-Stone theorem AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

A survey of algebraic extensions of commutative, unital normed algebras THOMAS DAWSON iii


99

135 145 151 157

CONTENTS

iv

Some more examples of subsets of c0 and £ 1 [0, 1] failing the fixed point property P. N. DOWLING,

C.

J. LENNARD, AND

B.

TURETT

Homotopic composition operators on H 00 ( Bn) PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

Characterization of conditional expectation in terms of positive projections J. J. GROBLER AND M. DE KocK

The Krull nature of locally C* -algebras MARINA HARALAMPIDOU

171 177 189 195

Characterizations and automatic linearity for ring homomorphisms on algebras of functions 0SAMU HATORI, TAKASHI ISHII, TAKESHI MIURA, AND SIN-EI TAKAHASI

Carleson embeddings for weighted Bergman spaces HANS JARCHOW AND URS KOLLBRUNNER

Weak* -extreme points of injective tensor product spaces KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO Determining sets and fixed points for holomorphic endomorphisms KANG-TAE KIM AND STEVEN G. KRANTZ

Localization in the spectral theory of operators on Banach spaces T. L. MILLER,

v.

G. MILLER, AND M. M. NEUMANN

Abstract harmonic analysis, homological algebra, and operator spaces VOLKER RUNDE

Relative tensor products for modules over von Neumann algebras DAVID SHERMAN

Uniform algebras generated by unimodular functions STUART J. SIDNEY

Analytic functions on compact groups and their applications to almost periodic functions THOMAS TONEY AND S. A. GRIGORYAN


201 217 231 239 247 263 275 293

299

Preface The Fourth Conference on Function Spaces was held at Southern Illinois University at Edwardsville from May 14 to May 19, 2002. It was attended by over 100 participants from 25 countries. The lectures covered a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), £P-spaces, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces, and other related subjects. The main purpose of the conference was to bring together mathematicians interested in various problems within the general area of function spaces and to allow a free discussion and exchange of ideas with people working on exactly the same problems as well as with people working on related questions. Hence, most of the lectures, and therefore the papers in this volume, have been directed to non-experts. A number of articles contain an exposition of known results (known to experts) and open problems; other articles contain new discoveries that are presented in a way that should be accessible also to mathematicians working in different areas of function spaces. The conference was the fourth in a sequence of conferences on function spaces at SlUE, with the first held in the spring of 1990, the second in the spring of 1994, and the third one in the spring of 1998. The proceedings of the first two conferences were published with Marcel Dekker in the Lecture Notes in Pure and Applied Mathematics series ( #136 and #172); the proceedings of the third conference were published by the AMS in the Contemporary Mathematics series (#232). The abstracts, the schedule of the talks, and other information, as well as the pictures of the participants, are available on the conference Web page at http://www.siue.edu/MATH/conference/. The conference was sponsored by grants from Southern Illinois University and from the National Science Foundation. The editor would like to thank everyone who contributed to the proceedings: the authors, the referees, the sponsoring institutions, and the American Mathematical Society. The editor would also like to express very special thanks to his wife, Dorota, for her active professional help during all of the stages of the organization - without her help the conference and the proceedings would not have been possible. Krzysztof Jarosz

v


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http://dx.doi.org/10.1090/conm/328/05765 Contemporary Mathematics Volume 328, 2003

Components of resolvent sets and local spectral theory Pietro Aiena and Fernando Villafane ABSTRACT. In this paper we shall study the components of various resolvent sets associated with some spectra originating from Fredholm theory. In particular, we obtain a classification of these components by using, in the case of operators of Kato type, the equivalences between the single valued extension property at a point and some kernel-type and range type conditions established in [2], [6], [3] and [5]. We also show that certain subspace valued mappings coincide on the components of the Kato type resolvent and give a precise description of the operators having empty Kato type spectrum.

1. Single valued extension property

Throughout this paper, T is assumed to be a bounded linear operator on a complex Banach space X and L(X) will denote the algebra of all bounded linear operators on L( X). If x E X, the local resolvent set of T at x E X, denoted by pr(x), is defined as the union of all open subsets U of ..I - T) f ()...)

=x

for all )... E U .

The local spectrum ar(x) ofT at x is defined by ar(x) := .. 0 , the only analytic function f : U ---+ X which satisfies the equation (U- T)f(>..) = 0 for all )... E U is the function f = 0. The operator T E L(X) is said to have SVEP if T has SVEP at every point)... E .. 0 , then the analytic solution of (1.1) in a neighborhood U of >.. 0 is uniquely determined. The SVEP was first introduced by Dunford [8], [9] and has later received a systematic treatment in Dunford-Schwartz [10]. The SVEP at a point was first introduced by Finch [11] and successively investigated by several authors, see [20], [28], [2], [6], [3] and [5] . The basic role of SVEP arises in local spectral theory, since every operator which satisfies the so-called Bishop's property ((3) enjoys this 1991 Mathematics Subject Classification. Primary 47A10, 47All. Secondary 47A53, 47A55. Key words and phrases. Single valued extension property, semi-regular operators, Kato decomposition property . The research was supported by the International Cooperation Project between the University of Palermo (Italy) and the University of Barquisimeto. @ 2003 American Mathematical Society


2

PIETRO AlENA AND FERNANDO VILLAFANE

property, see [18] for definition and results. Recall that an operator T E L(X) on a Banach space X is said to be decomposable if for every open cover {U1 , U2} of C there exist T-invariant closed linear subspaces X 1 and X 2 of X for which X = xl + x2, cr(T lXI) ~ ul and cr(T IX2) ~ U2. The class of decomposable operators contains, for instance, all normal operators, all spectral operators, all operators with a non-analytic functional calculus and all compact operators, or more generally all operators with a totally disconnected spectrum. Note that T is decomposable if and only if T and its dual T* have property (;3), see Theorem 2.5.19 [18]. Consequently, if Tis decomposable, then both T and T* have SVEP. For an arbitrary subset F of C let Xr(F) be the local spectral subspace associated with F, defined by Xr(F) := {x EX: crr(x) ~ F}. IfF is a closed subset of C, let Xr(F) be the glocal spectral subspace associated with F, defined as the set of all x E X for which there exists an analytic function f : C \ F ____, X which satisfies (>..I- T)f(>..) = x for all>.. E C \F. Clearly, Xr(F) and Xr(F) are (not necessarily closed) linear subspaces of X with Xr(F) ~ Xr(F) for all closed sets F ~ C. Note that, by Proposition 3.3.2 of [18], the identity Xr(F) = Xr(F) holds for all closed sets F ~ C precisely when T has SVEP, and this is the case if and only if Xr(0) = {0}, see Proposition 1.2.16 of [18]. The SVEP at a point >..0 may be characterized in a similar way: T has SVEP at >.. 0 if and only if ker (>.. 0 ! - T) n Xr(0) = {0}, cf. [1, Theorem 1.9]. Two important subspaces in Fredholm theory are the hyperrange of T, defined by r=(X) = n~=l Tn(X), and the hyperkernel ofT defined by N=(T) = U~=l ker yn. Recall that the ascent of an operator T is the smallest non-negative integer p := p(T) such that ker TP = ker TP+l. If such integer does not exist, we put p(T) = oo. Obviously, if T has finite ascent p then N 00 (T) = ker TP. Analogously, the descent q := q(T) of an operator T is the smallest non-negative integer q such that Tq(X) = yq+ 1 (X). If such integer does not exist we put q(T) = oo. Also, if T has finite descent q then r=(x) = Tq(X). It is well-known that if p(T) and q(T) are both finite then p(T) = q(T), see [15, Proposition 38.3]. Furthermore, p(>.. 0 I- T) = q(>.. 0 I- T) < oo if and only if >.. 0 is a pole of the resolvent R(>..,T) :=(>..I -T)- 1 , [15, Proposition 50.23]. Associated with T E L(X) there is another linear subspace of X, the quasinilpotent part of T defined as

Ho(T)

:=

{x EX: lim IITnxlll/n n--->oo

=

0}.

Evidently, N 00 (T) ~ H 0 (T). Moreover, H 0 (T) = X if and only if T is quasinilpotent, i.e. cr(T) = {0}, see [20, Theorem 1.5]. The following decomposition property studied by Mbekhta [20], [19], Mbekhta and Ouahab [21], has its origin in the classical treatment of perturbation theory due to Kato [17], who showed an important decomposition for semi-Fredholm operators. DEFINITION 1.1. An operator T E L(X) is called semi-regular if T(X) is closed and N 00 (T) ~ T 00 (X). An operator T E L(X) is said to be of Kato type if there exists a pair ofT-invariant closed subspaces (M, N) such that X = M EB N, the restriction T IM is semi-regular and T IN is nilpotent. The pair (M, N) is called a generalized Kato decomposition ( GKD, for brevity ) forT.


COMPONENTS OF RESOLVENT SETS AND LOCAL SPECTRAL THEORY

3

If, additionally, in the definition above we assume that N is finite-dimensional then T is said to be essentially semi-regular, see Rakocevic [25] and Muller [24]. By Proposition 3.1.6 of [18], T is semi-regular if and only if T* is semi-regular. Analogously, by Corollary 3.4 of [24], T is essentially semi-regular if and only if T* is essentially semi-regular. Moreover, if Tis essentially semi-regular then rn is essentially semi-regular for all n EN, again by Corollary 3.4 of [24]. It should be noted that the range of an essentially semi-regular operator is always closed. In fact, if (M, N) is a GKD for T with N finite-dimensional, then T(X) is the direct sum of T(M), which is closed because T I M is semi-regular, and of T(N), which is finite-dimensional. Two very important class of essentially semi-regular operators is given by the class of upper semi-Fredholm operators, defined by

+(X) := {T E L(X) :dim ker T < oo, T(X) is closed} and the class of all lower semi-Fredholm operators defined by

_(X)

:= {T E

L(X): codim T(X) < oo},

see Kato [17, Theorem 4] or West [31]. The class of Fredholm operators is defined by (X) := +(X)n_(X). Note that a semi-Fredholm operator Tis semi-regular if and only if its jump j(T) is zero, see ([31, Proposition 2.2]. Moreover, if T is of Kato type, and in particular if T is semi-regular, then T 00 (X) is closed with T(T 00 (X)) = T 00 (X), see Theorem 2.3 and Theorem 2.4 of [2], and T 00 (X) coincides with the analytical core K(T) := {x EX: there exist a constant c > 0 and a sequence of elements Xn EX such that x 0 = x, Txn = Xn_ 1 , and llxnll :::; cnllxll for all n E N}. It should be noted that both subspaces K(T) and H 0 (T) admit a local spectral characterization. In fact,

= Xr(C \ {0}) = {x EX: 0 ~ crr(x)}, see Mbekhta [20], Vrbova [30] and also Propositions 3.3.7 and 3.3.13 of [18], and H 0 (T) = Xr( {0}, see Propositions 3.3.7 and 3.3.13 of [18]. Therefore, if T has SVEP, then H 0 (T) = Xr( {0} ). (1.2)

K(T)

In the following lemma by A.L and its proof we denote the annihilator of a subset A ~ X, and by .L B the pre-annihilator of a subset B ~ X*. LEMMA

1.2. For every T

(i) H 0 (T)

E

L(X), the following statements hold

~j_ K(T*) and K(T) ~j_

H 0 (T*).

(ii) If T is a Kato type operator and the pair (M, N) is a GKD forT then

K(T) = K(T I M) = K(T) n M, and

(1.3)

Ho(T) = Ho(T I M) EB Ha(T I N) = Ho(T I M) EB N.

(iii) If T is essentially semi-regular then (1.4)

H 0 (T)

=

N 00 (T)

=.L

(iv) IfT is semi-regular then H 0 (T)

K(T*) and K(T) ~

=.L

H 0 (T*),

K(T).



4

PROOF. (i) See Proposition 4.1 of [5].

(ii) The proof of the equality K(T) = K(T I M) may be found in [1]. The second equality in (1.3) is clear, since the nilpotency ofT I N implies H 0 (T I N) = N. The inclusion H 0 (T I M) + H 0 (T I N) ~ H 0 (T) is evident. To show the opposite inclusion, consider an arbitrary element x E H 0 (T) and set x = y + z, with y E M, z E N. Since T I N is quasi-nilpotent then N = H 0 (T) ~ H 0 (T). Therefore y = x- z E H 0 (T) n M = Ho(T I M) and consequently Ho(T) ~ Ho(T I

M)

+ Ho(T

IN).

(iii) Assume that T is essentially semi-regular and hence also T* essentially semi-regular. Then T*n is essentially semi-regular for all n E N, so that T*n(X*) is closed for all n EN. From part (i) we know that N 00 (T) ~ H 0 (T) ~J. K(T*), so that, to show the first two equalities of (1.4), we need only to prove the inclusion J. K(T*) ~ N 00 (T). For every T E L(X) and every n EN we have ker rn ~ N 00 (T) and hence N 00 (T)J. ~ ker rn.l. = T*n(X*), because the last subspace is closed for ~---,--,-.1.

all n EN. From this we easily obtain that N 00 (T) ~ T* 00 (X*) = K(T*), where the last equality holds since T* is essentially semi-regular and hence of Kato type. Consequently, J.K(T*) ~ N 00 (T). Thus the first two equalities of (1.4) are proved. The equality K(T) =J. H 0 (T*) is proved in a similar way. (iv) The semi-regularity ofT entails that N 00 (T) ~ T 00 (X) = K(T), so that, by part (ii), H 0 (T) = N 00 (T) ~ K(T) = K(T) , and this concludes the proof. D We have already observed that an operator T E L(X) has SVEP at >. 0 precisely when ker (>. 0 ! - T) n K(>. 0 I- T) = {0}. From the inclusion ker (>.of-T)~

N 00 (Aof- T)

~

Ho(.Aoi- T)

it then follows that the condition H 0 (>. 0 I- T) n K(>. 0 I- T) = {0} implies that T has SVEP at >. 0 . Example 2.5 of [5] shows that SVEP for T at a point does not necessarily imply that H 0 (>. 0 I- T) n K(>. 0 I- T) = {0}. In [2] it has been shown that also the condition N 00 (.A 0 I- T) n (>. 0 !- T) 00 (X) = {0} implies SVEP at >. 0 for T. The next result shows that these implications are actually equivalences in the case that >. 0 ! - T is of Kato type. THEOREM 1.3. If >. 0 ! - T is of Kato type then the following properties are equivalent: (i) T has the SVEP at >. 0 ; (ii) Ho(.A 0 I- T) n K(.Aoi- T) = {0}; (iii) H 0 (>. 0 I- T) is closed; (iv) >. 0 ! - T has finite ascent; (v) N 00 (.A 0 I- T) n (>. 0 ! - T) 00 (X) = {0}. Furthermore, if >. 0 ! - T E L(X) is essentially semi-regular, the assertions (i)(v) are equivalent to the following conditions: (vi) H 0 (>. 0 I- T) is finite-dimensional. (vii) N 00 (.A 0 I- T*) + (>. 0 ! - T*)(X*) is weak *-dense in X*; (viii) H 0 (>. 0 I- T*) + (>. 0 ! - T*)(X*) is weak *-dense in X*; (ix) Ho(>. 0 I- T*) + K(>. 0 I- T*) is weak *-dense in X*. In this case >. 0 ! - T E .. 0 1- Tis essentially semi-regular. We may assume that >.. 0 = 0. (i) .;=? (vi) Obviously, if H 0 (T) is finite-dimensional then H 0 (T) is closed, soT has SVEP at 0, by the equivalence (i) .;=? (iii). Conversely, if T has SVEP at 0 then also T I M has SVEP at 0, since the local SVEP is inherited by the restrictions to closed invariant subspaces. The semiregularity ofT I M then implies that T I M is injective, see Theorem 2.14 of [1] and therefore N 00 (T) = {0}. By part (iii) of Lemma 1.2 we then conclude that H 0 (T I M) = Noo(T I M) = {0}. From part (ii) of Lemma 1.2 it follows that H 0 (T) = {0} EB N = N is finite-dimensional. Finally, if T is essentially semi-regular then also rn is essentially semi-regular and therefore, the ranges rn(X) are closed for all n E N. From Theorem 4.3 of [5] it then follows that the condition (i) is equivalent to each one of the conditions (vii), (viii) and (ix). It remains to establish that (i) implies that >.. 0 1- T E .. 0 I - T) is finite- dimensional then also its subspace ker (>.. 0 1 - T) is finitedimensional. Since ( >.. 0 1- T) (X) is closed we then conclude that >.. 0 1- T E + (X). D The next Theorem 2.1 will show that, if >.. 0 1- T of Kato type, then H 0 (>.. 0 IT) n K(>..oi- T) = N 00 (A.oi- T) n (>..oJ- T) 00 (X). The following characterizations of SVEP for the dual T* are dual, in a sense, to those given in Theorem 1.3. THEOREM 1.4. Suppose that >.. 0 1 - T is of Kato type. Then the following statements are equivalent: (i) T* has SVEP at A.o; (ii) X= Ho(A.oi- T) + K(>..oi- T); (iii) >.. 0 1 - T has finite descent; (iv) X= N 00 (A.oi- T) + (>..oi- T) 00 (X); (v) Ho(>..oi- T) + K(>..oi- T) is norm-dense in X; (vi) N 00 (>.. 0 J- T) + (>.. 0 1- T) 00 (X) is norm-dense in X; Furthermore, if >.. 0 1- T E L(X) is a essentially semi-regular then the assertions (i)-(vi) are equivalent to the following cond,itions: (vii) K(>.. 0 I- T) is finite-codimensional; (viii) Noo (>.. 0 1 - T) + (>.. 0 1 - T) (X) is norm-dense in X; (ix) H 0 (>.. 0 I- T) + (>.. 0 1- T)(X) is norm-dense in X. In this case >.. 0 1 - T E _ (T). PROOF. Also here we may assume that >.. 0 = 0 and T is of Kato type. The equivalence of (i), (ii) and (iii) has been established in Theorem 2.9 of [3]. The equivalence of (i) and (iv) has been proved in Theorem 2.9 of [2], see also Theorem 2.1 of the present paper. Clearly, (ii) =} (v), (iv) =} (vi). The implications (v) =} (i) and (vi) =} (i) have been proved in Corollary 4.2 of [5], so that the statements (i)-(vi) are equivalent. Now, assume that T is essentially semi-regular. Then rn(X) is closed for



6

all n E N, so that, by Theorem 4.3 of [5], the statements (i), (viii) and (ix) are equivalent. To conclude the proof note first that if (M, N) is a GKD forT then the pair (Nl.,Ml.) is a GKD forT*. Now, if T* has SVEP at 0, then, as observed in the proof Theorem 1.3, T* I N l. is injective and therefore, see Lemma 2.8 of [2], T I M is surjective. Therefore K(T) = K(T I M) = M is finite-codimensional, so that the implication (i) =? (vii) is proved. Conversely, suppose that the analytical core K(T) is finite-codimensional. From K(T) = T 00 (X) ~ Tn(X) we deduce that q(T) < oo, so that (vii) implies (iii), and the proof of the equivalences is complete . Finally, from the inclusion K(>. 0 I -T) ~ (>. 0 1 -T)(X) we infer that, if K(>. 0 IT) finite-codimensional, then also (>. 0 1 - T)(X) is finite-codimensional, so that >. 01- T E _(X). 0 Recall that T E L(X) is called bounded below if T is injective and has closed range T(X). It is easily seen from definition of SVEP that, if the approximate point spectrum aap(T) := {>.. E C: >.I-T is not bounded below} does not cluster at >..0 , then T has SVEP at >.. 0 and, dually, if the surjectivity spectrum a su (T) = {>.. E C : >.I - T is not surjective} does not cluster at >..o, then T* has SVEP at >..o. The next result shows that for Kato type operators these implications may be reversed. THEOREM

1.5. If >. 0 1- T is of Kato type, then the following equivalences hold:

(i) T has the SVEP at >.0 precisely when aap(T) does not cluster at >.0 , [6, Theorem 2.2]; (ii) T* has the SVEP at >.o precisely when a 8 u(T) does not cluster at >.o, [6, Theorem 2.5]. 2. Components In this section we shall take a closer look at the components of some resolvent sets associated with the various spectra originating from Fredholm theory. In particular, we shall obtain a classification of these components, by using the constancy of some mappings and the equivalences between the SVEP at a point and the kernel-type and range type conditions, established in the previous section. For an operator T E L(X), we consider the following parts of the ordinary spectrum: the Kato spectrum ak(T) := {>.. E C: >.I-T is not semi-regular},

and the essential Kato spectrum ake(T) := {>.. E C: >.I-T is not essentially semi-regular}.

Moreover, we define O'kt(T) := {>.. E C: >.I-T is not of Kato type}.



7

It is known that the three sets CJk(T), CJkt(T) and CJke(T) are closed, for the first set see [18, Proposition 3.1.9], for the other two sets see [4, Corollary 1]. Moreover, CJk(T) and CJke(T) are nonempty, since the first spectrum contains the boundary of CJ(T), see [18, Proposition 3.1.6], while the second spectrum contains the boundary of the Fredholm spectrum CJ 1 (T) := {-\ E C: ,\I-T rf. (X)}, see [24, Theorem 3.8]. Next we shall show that CJkt(T) is non-empty precisely when CJ(T) is not a finite set of poles. Let Pk(T) := C \ CJk(T), Pkt(T) := C \ (Jkt(T) and Pke(T) := C \ CJke(T) be the resolvents associated with these spectra. The sets Pk(T), Pkt(T) and Pke(T) are open subsets of C, so they may be decomposed in connected disjoint open non-empty components. Clearly,

(2.1)

Pk(T) ~ Pke(T) ~ Pkt(T).

Note that for every T E L(X) we have Pk(T) In [29]

6

= Pk(T*) and Pke(T) = Pke(T*).

Searcoid and West showed the constancy of the mappings

on the components of the semi-Fredholm resolvent Pst(T) := C \ CJ 8 j(T), where CJ 8 j(T) is the semi-Fredholm spectrum defined by

CJ 8 J(T) := {-\ E C: -\I-T

rf.

+(X) U _(X)}.

From the Kato decomposition for semi-Fredholm operators we easily obtain the following inclusions (2.3)

Pst(T) ~ Pke(T) ~ Pkt(T).

The work of 6 Searcoid and West [29] extended previous results established by Homer [16], by Goldmann and Krackovskii [13], [14], and by Saphar [26], which have established the constancy of the functions on a component of the semi-Fredholm resolvent Pst(T), except for the discrete subset of points for which -\I-T is not semi-regular. In the same vein, Forster [12] showed that the mappings

,\---. K(AI- T) = (AI- T) 00 (X),

,\---. Ho(AI- T) = N=(AI- T)

are constant as,\ ranges through a component of the Kato resolvent Pk(T). The constancy of these mappings has also been studied by Mbekhta and Ouahab [22], which showed the constancy of the mappings (2.4)

,\---. H 0 (AI- T)

+ K(AI- T),

,\---. H 0 (AI- T) n K(AI- T)

on the components of Pkt(T). The next result shows that the mappings (2.2) and (2.4) coincide, respectively, on the components of Pkt(T), so that the Mbekhta and Ouahab result extends the previous result of 6 Searcoid and West. 2.1. Let -\I-T be of Kato type. Then (i) N=(-\I- T) +(-\I- T)=(X) = H 0 (AI- T) + K(-\I- T).

THEOREM

(ii) N=(AI- T) n (AI- T)=(X) = H 0 (AI- T) n K(-\I- T).



8

PROOF. (i) Throughout this proof we may take A= 0. Let (M, N) be a GKD forT such that (T I N)d = 0 for some integer dEN. By part (ii) of Lemma 1.2 we know that K(T) = K(T I M) = K(T) n M. Moreover, by part (iv) of Lemma 1.2, the semi-regularity ofT I M implies that H 0 (T I M) ~ K(T I M) = K(T). From this we obtain

= H 0 (T) n (K(T) n M) = (H0 (T) n M) n K(T) = Ho(T I M) n K(T) = Ho(T I M),

H 0 (T) n K(T)

and therefore Ho(T) n K(T) = Ho(T I M). We claim that H 0 (T) + K(T) = NEB K(T). From N ~ ker Td ~ H 0 (T) we obtain that NEB K(T) ~ H 0 (T) + K(T). Conversely, from part (ii) of Lemma 1.2 we have

Ho(T) =NEB H 0 (T I M) =NEB (H0 (T) n K(T)) ~NEB so that

H 0 (T)

+ K(T)

~(NEB

K(T))

+ K(T)

K(T),

K(T),

~NEB

so our claim is proved. Since K(T) = T 00 (X) for every operator of Kato type, we obtain from the inclusion N ~ ker Td ~ N 00 (T), that

H 0 (T)

+ K(T)

so the equality N

00

=NEB K(T) ~ N

(T)

+T

00

00

(X) = H 0 (T)

(T)

+T

+ K(T)

00

(X) ~ H 0 (T)

+ K(T),

is proved.

(ii) Suppose again that A= 0. Let (M, N) be a GKD forT such that, for some

d E N, we have (T IN)d = 0. Then ker yn = ker (T IM)n for every natural n Since ker yn ~ ker yn+ 1 for all n E N we then have

N 00 (T) =

Uker yn = Uker(T I M)n = N 00

00

n?_d

n?_d

00

2: d.

(T I M).

The semi-regularity ofT I M then implies, by part (iii) of Lemma 1.2, that (2.5)

Noo(T) = Noo(T I M) = Ho(T I M) = Ho(T) n M.

Next we show that the equality H 0 (T) n M = H 0 (T) n M holds. The inclusion H 0 (T) n M ~ H 0 (T) n M is evident. Conversely, suppose that x E H 0 (T) n M. Then there is a sequence (xn) C H 0 (T) such that Xn ---> x as n---> oo. Let P be the projection of X onto M along N. Then Pxn ---> Px = x and Pxn E Ho(T) n M. Therefore X E Ho(T) n M. Finally, from (2.5) and taking into account that K(T) n M = K(T) = T 00 (X), we then obtain

N

00

(T) n T 00 (X) = H 0 (T) n M n K(T) = H 0 (T) n (M n K(T)) = H 0 (T) n K(T),

so the proof is complete.

D

From the constancy of the mappings A---> H 0 (AI- T)nK(AI -T), or, which is the same, of the mappings A---> N 00 (Al- T) n (.AI- T)=(X), on the components of Pkt (T) and the results established in the previous section we now obtain the following classification.



9

THEOREM 2.2. LetT E L(X) and n a component of Pkt(T). Then the following alternative holds: either (i) T has SVEP for every point of n. In this case p()..I- T) < oo for all).. E n. Moreover, (Jap(T) does not have limit points in !l; every point of!l, except possibly for at most countably many isolated points, is not an eigenvalue ofT. or (ii) T has SVEP at no point of n. In this case p(U- T) = oo for all ).. E n. Every point of n is an eigenvalue ofT, PROOF. (i) Suppose that T has SVEP at Ao E !1. Then, by Theorem 1.3, H 0 ()..J- T) is closed and

H 0 (Aoi- T) n K(Aoi- T)

= Ho(Aoi- T) n K(Aoi- T) = {0}.

Since the mapping ).. ----+ H 0 ()..I- T) n K()..I- T) is constant on the component n, then H 0 (U- T) n K(U- T) = {0} for all ).. E n and this implies, again by Theorem 1.3, that T has SVEP at every ).. E n. This is equivalent, also by Theorem 1.3, to saying that p(AI- T) < oo for all ).. E n. Moreover, from Theorem 1.5, (Jap(T) does not cluster inn and, consequently, every point of n is not an eigenvalue ofT, except a subset of n which consists of at most countably many isolated points. (ii) This is clear, again by Theorem 1.3.

0

Recall that ).. E C is said to be a deficiency value for if )..I- T is not surjective. THEOREM 2.3. LetT E L(X) and n a component of Pkt(T). Then the following alternative holds: either (i) T* has SVEP for every point of n. In this case q(AI - T) < oo for all ).. En. Moreover, (J 8 u(T) does not have limit points in !l; every point of !1, except possibly for at most countably many isolated points, is not a deficiency value ofT. or (ii) T* has the SVEP at no point of n. In this case q(U- T) = oo for all ).. E !1 and every).. E !1 is a deficiency value ofT. PROOF. Proceed as in the proof of Theorem 2.2, combining the constancy on the components of Pkt(T) of the mapping).. En----+ K(U- T) + H 0 ()..J- T) (or, equivalently, the constancy of the mapping).. E n ----+ N 00 ()..J- T) + (U- T) 00 (X) ), with Theorem 1.4 and Theorem 1.5. 0 The previous results lead to a precise description of the operators whose Kato type spectrum (Jkt (T) is empty. Most of the results of the following theorem may be found in Mbekhta [23] in the context of operators on Hilbert spaces. However, our proofs, involving local spectral theory, are considerably simpler and are established in the more general context of operators acting on Banach spaces. Recall first that T E L(X) is algebraic if there exists a non-trivial polynomial h such that h(T) = 0. THEOREM 2.4. For an operator T E L(X) the following statements are equivalent: (i) (Jkt(T) is empty;


10

PIETRO AlENA AND FERNANDO VILLAFANE (ii) )..1- T has finite descent for every ,\ E .I-T has finite descent for every ,\ E au (T), where au (T) is the topological boundary of u(T); (iv) u(T) is a finite set of poles of R(>., T);

(v) T is algebraic. PROOF. (i) =? (ii) Suppose that Ukt(T) = 0. Then Pkt(T) has an unique component 0 = .I- T) = P(X), see the proof of Theorem 1.6 of [19] or also Theorem 1 of [27]. Since ,\ is a pole of R(>., T), by Proposition 50.2 of [15], >.I - T has positive finite ascent and descent, and if p := p(>. 0 I - T) = q(AI- T), then N = ker (AI- T)P. From the classical Riesz functional calculus we know that u(T I M) = u(T) \ {,\}, [15, Theorem 49.1], so that (AI- T) I M is bijective, while (AI- T I N)P = 0. Therefore >.I-T is of Kato type for every ,\ E .) := (-\ 1 - ,\)P' ···(An- ,\)Pn.

Then, see Lemma 3.1.15 of

[18],

h(T)(X) =

n

n

i=1

i=1

n(>-ii- T)Pi(X) = n

K(>.J- T),

where the last equality follows since T has SVEP and Ail - T is of Kato type, see Theorem 2.9 of [3]. But the last intersection is {0}, since, by the local spectral characterization of the analytical core (1.2), if x E K(>.J- T) n K(>.ji- T), with Ai -=1- Aj, then ur(x) ~ {Ai} n {>.j} = 0 and hence x = 0, since T has SVEP. Therefore h(T) = 0. (v) =? (i) As in the proof of (iv) =? (i) it suffices to show that AI - T is of Kato type for all,\ E u(T). Let h be a polynomial such that h(T) = 0. From the spectral mapping theorem we easily deduce that u(T) is a finite set {,\ 1 , · · · , An}· The points -\ 1 , · · · , An are zeros of finite multiplicities of h, say k 1 , · · · , kn, respectively, so that h(>.) = (-\1 - >.)k 1 • • • (>-n - >.)kn and hence

X= ker h(T) =

n

E9 ker (>.J- T)k', i=l



11

see Lemma 3.1.15 of [18]. Now, suppose that >. = >.1 for some j and define ho(>.) :=

IJ (>.i- >.)k'. iojj

We have

M := ker h 0 (T) =

E9 ker (>.J- T)k' if.j

and if N := ker (>. 1 I - T)kJ, then X = M E9 N and M, N are invariant under >.1 I - T. From the inclusion ker (>. 1 I - T) t:;; ker (>. 1 I - T)kJ = N, we infer that the restriction of >.1 I - T on M is injective. It is easily seen that (>. 1 I- T)(ker (>.ii- T)ki) = ker (>.J- T)k,,

i

=1 j,

so that ( >.1I-T) ( M) = M. Hence the restriction of >.1I-T on M is also surjective and therefore bijective. Obviously, (>. 1 I- T) I N)kJ = 0, so that >.1 I- Tis of Kato type, as desired. 0 A bounded operator on a Banach space X is said to satisfy a polynomial growth condition, if there exists a K > 0, a > 0 for which

o

llexp(i>.T)II :::; K(1

+ 1>-1 8 )

for all>. E ffi.,

Examples of operators which satisfy a polynomial growth condition are hermitian operators on Hilbert spaces, nilpotent and projection operators, algebraic operators with real spectra, see Barnes [7]. In Laursen and Neumann [18, Theorem 1.5.19] it is shown that the class P(X) of operators which satisfy a polynomial growth condition coincides with the class of all generalized scalar operators having real spectra. As noted in Barnes [7], if T E P(X) and >. 0 I - T has closed range for some >. 0 E C then q(>. 0 I - T) is finite. From Theorem 2.4 it follows that, if T E P(X), then the condition (>.!- T)(X) closed for all >. E C implies that C!kt (T) = 0. Other classes of operators for which CJkt (T) = 0 may be found in [23]. The classification of the components of Pes (T) may be easily obtained from Theorem 2.2 and Theorem 2.3, once it has been observed, that the two sets Pes(T) and Pkt(T) may be different only for a denumerable set, see for instance Corollary 1 of [4]. We now look at the components of Pst(T). Recall that for a semi-Fredholm operator T E cll+(X) U cll_(X) we can consider the index defined by ind T := dim ker T- codim T(X). Clearly, from Theorem 2.2 and Theorem 2.3, T, as well as T*, has SVEP either for every point or no point of a component 0 of Pst(T). We can classify the components of Pst(T) as follows: THEOREM 2.5. LetT E L(X) and 0 a component of Pst(T). For the SVEP, the index, the ascent and the descent on 0 there are exactly the following four possibilities:

(i) Both T and T* have SVEP at every point of 0. In this case we have ind (>.I-T) = 0 and p(>.I- T) = q(>.I- T) < oo for every>. E 0. The eigenvalues and deficiency values do not have a limit point in 0. This case occurs exactly when 0 intersects the resolvent p(T). (ii) T has SVEP at the points of 0, while T* fails to have SVEP at the points of 0. In this case we have ind (>.!- T) < 0, p(>.I- T) < oo and q(>.I- T) = oo


12


for every A E fl. The eigenvalues do not have a limit point in f2 and every point of n is a deficiency value. (iii) T* has SVEP at the points of 0, while T fails to have SVEP at the points of fl. In this case we have ind (AI- T) > 0, p(AI- T) = oo and q(Al- T) < oo for every A E fl. The deficiency values do not have a limit point in 0, while every point of n is an eigenvalue. (iv) Neither T or T* has SVEP at the points of fl. In this case we have p(AI- T) = q(AI- T) = oo for every A E fl. The index may assume every value in 7!.,; all the points of n are eigenvalues and deficiency values. PROOF. The case (i) is clear from the results established in the previous section, Theorem 2.2 and Theorem 2.3. The index ind (A/- T) = 0 by Proposition 38.6 of [15]. In the case (ii) the condition p( AI- T) < oo implies that AI- T has index less or equal to 0, while the condition q(AI- T) = oo excludes that ind (AI- T) = 0, see Proposition 38.5 of [15]. A similar argument shows in the case (iii) that ind (AI- T) > 0. The statements of (iv) are clear. 0 The following corollary establishes that a very simple classification of the components ofsemi-Fredholm resolvent may be obtained in the case that T, or T* has SVEP. Recall that the case that both T and T* have SVEP applies in particular to the decomposable operators. COROLLARY 2.6. LetT E L(X) and f2 any component of Pst(T). If T has SVEP then only the case (i) and (ii) of Theorem 2.5 are possible, while if T* has SVEP only the case (i) and (iii) are possible. Finally, if both T and T* have SVEP then only the case (i) is possible. In the next result we consider the components of Pk(T), which is the smallest of the resolvent sets that we have considered. THEOREM 2.7. LetT E L(X) and f2 any component of Pk(T). Then one of the following possibilities occurs: (i) Both T and T* have SVEP at every point of fl. In this case we have n ~ p(T). (ii) T has SVEP at the points of 0, while T* fails to have SVEP at every point ofrl. In this case we have f2 n aap(T) = 0 and f2 ~ a 8 u(T). (iii) T* has SVEP at the points ofrl, while T* fails to have SVEP at the points of fl. In this case we have f2 n asu(T) = 0 and f2 ~ aap(T). (iv) Neither T or T* have SVEP at the points of fl. In this case we have f2 ~ aap(T) n asu(T). PROOF. (i) Let Ao E fl. The subspaces M :=X and N := {0} give a GKD for T and the subspaces .l N =X and .l M = {0} give a GKD forT*. As observed in the proof of Theorem 1.3 and Theorem 1.4 if T has SVEP at Ao then Ho(Aol- T) = N = {0} and if T* has SVEP at Ao then K(Aol- T) = M = X. Therefore Ao E p(T). (ii) In this case H 0 (>.I- T) = {0} and (AI- T)(X) is closed for every A E fl. If A ~ a 8 u(T) then A E p(T) = p(T*) and this is impossible, since T* does not have SVEP at A.



13

(iii) In this case K(>J- T) =X for every A E S1, so A.A gehor-en. Arch. Math. 17 (1966), 56-64. [13] M. A. Goldman, S. N. Krackovskii InvaTiance of cer-tain subspaces associated with A- >.I. Soviet Math. Doklady 5, (1964), 102-4. [14] M. A. Goldman, S. N. Krackovskii Behaviour- of the space of zero elements with finitedimensional salient on the zero ker-nel under pertur-bations of the operator. Soviet Nath. Doklady 16, (1975), 370-3. [15] H. Heuser Functional Analysis (1982), Marcel Dekker, New York. [16] R. H. Homer Regular- extensions and the solvability of operator- equations. Proc. Amer. Math. Soc. 12 (1961), 415-18. [17] T. Kato Per-tur-bation theory for nullity, deficiency and other quantities of linear- opemtor-s. J. Anal. Math. 6 (1958), 261-322. [18] K. B. Laursen, M. M. Neumann Introduction to local spectral theory, Clarendon Press, Oxford 2000. [19] M. Mbekhta Sur- l'unicite de la decomposition de Kato generalisee. Acta Sci. Math. (Szeged) 54 (1990), 367-77. [20] M. Mbekhta Sur- la theor-ie spectrale locale et limite des nilpotents. Proc. Amer. Math. Soc. 110 (1990), 621-631. [21] M. Mbekhta, A. Ouahab Operateur- s-regulier- dans un espace de Banach et theoTie spectrale. Acta Sci. Math. (Szeged) 59 (1994), 525-43. [22] M. Mbekhta, A. Ouahab Pertur-bation des operateurs s-r-egulier-s . Topics in operator theory, operator algebras and applications, Timisoara (1994), Rom. Acad. Bucharest, 239-249. [23] M. Mbekhta Ascent, descent et spectr-e essential quasi-Fr-edholm., Rendiconti Circ. Mat. Palermo (2), 46, (1997), 175-196. [24] V. Muller On the regular spectr-um., J. Operator Theory 31 (1994), 363-380. [25] V. Rakocevic Generalized spectr-um and commuting compact pertur-bation. Proc. Edinburgh Math. Soc. 36 (2), (1993), 197-209. [26] P. Saphar Contr-ibution a l'etude des applications lineaires dans un espace de Banach. Bull. Soc. Math. France 92 (1964), 363-84. [27] C. Schmoeger On isolated points of the spectrum of a bounded operator. Proc. Amer. Math. Soc. 117 (1993), 715-19.


14


[28] C. Schmoeger (1995). Semi-Fredholm opemtors and local spectml theory., Demonstratio Math. 4, 997-1004. [29] M. 6 Searc6id, T. T. West Continuity of the genemlized kernel and mnge for semi-Fredholm opemtors. Math. Proc. Camb. Phil. Soc. 105, (1989), 513-522. [30] P. Vrbova On local spectml properties of opemtors in Banach spaces. Czechoslovak Math. J. 23(98) (1973a), 483-92. [31] T. T. West A Riesz-Schauder theorem for semi-Fredholm opemtors. Proc. Roy. Irish. Acad. 87 A, N.2, (1987), 137-146. DIPARTMENTO DI MATEMATICA ED APPLICAZIONI, VIALE DELLE 8CIENZE, UNIVERSITA D1 PALERMO, 90128 PALERMO, ITALY E-mail address: [email protected] DEPARTAMENTO DE MATEMATICAS, FACULTAD DE CIENCIAS, UNIVERSIDAD UCLA DE BARQUISIMETO (VENEZUELA) E-mail address: fvillafa


The Fejer-Riesz Inequality and the Index of the Shift John R. Akeroyd ABSTRACT. In this brief article we consider a result that can be characterized as the converse of the Fejer-Riesz inequality. This result has bearing on theory concerning the index of the shift.

Let p, be a finite, positive Borel measure with support in { z : izl ~ 1} (][)) := { z : izl < 1}) and let P 2 (p,) denote the closure of the polynomials in L 2 (p,). We assume

throughout that P 2 (p,) is irreducible (i.e., it contains no nontrivial characteristic functions). From this it follows that: a) p,lallli « m (normalized Lebesgue measure on 8][))), and b) for any w in ][)), f ~--> f(w) defines a bounded linear functional for polynomials f with respect to the L 2 (p,) norm, that is bounded independent of w in any compact subset of][)); cf. [15], Theorem 5.8. In other words, ][)) = abpe(P 2 (p,)) -the collection of analytic bounded point evaluations for P 2 (p,). In the case that p,(8][))) > 0 and p,jllli is radially weighted area measure, there is much in the literature concerning which weights have the property that P 2 (p,) is irreducible; for instance, cf. [9] and [10]. Returning to our general setting, notice that multiplication by the independent variable z is a bounded operator on P 2 (p,). We call this operator the shift (on P 2 (p,)) and denote it by Mz, suppressing reference to p,. Let Lat(Mz) denote the collection of closed invariant subspaces for the shift (on P 2 (p,)). If {0} =f. M E Lat(Mz), then, since 0 E abpe(P 2 (p,)), zM is a closed subspace of M and in fact dim(M 8 zM) 2:: 1. In the case that p,(8][))) = 0, C. Apostol, H. Bercovici, C. Foias and C. Pearcy have shown that for any natural number n, and for n = oo, there exists Min Lat(Mz) such that dim(M8zM) = n; cf. [3], and for related work see [7]. This result is an indication of how very large Lat(Mz) is in the case that p,(8][))) = 0. In fact, it is large enough to "model" the general invariant subspace problem for bounded operators on a Hilbert space; again, cf. [3] and [7]. A classical example that falls under this heading (p,(8][))) = 0) is the Bergman space £~(][))), which equals P 2 (p,) when p, =A- area measure on ][)). At the other extreme, if p, = m, then P 2 (p,) represents the Hardy space H 2 (][))) and so, by Beurling's Theorem, dim(M 8 zM) = 1 for all nontrivial members M 1991 Mathematics Subject Classification. Primary 47A53, 47B20, 47B38; Secondary 30E10, 46E15.

15

©

2003 American Mathematical Society


JOHN R. AKEROYD

16

of Lat(A1z). It has been conjectured that for any measure fl with mass on the unit circle (i.e., M(8][)l) > 0), the outcome mimics that of Hardy space case and dim(M 8 zM) = 1 whenever {0} -=1-M E Lat(Mz); cf. [5]. There are a number of results in the literature that support this conjecture. The first of these is found in a paper of R. Olin and J. Thomson (cf. [12]) who show that it holds whenever fl that has a so-called "outer hole" in its support. Subsequently (in [11]), L. Miller shows that it also holds in the case that fl = A+ mil', where 1 is some nontrivial subarc of 8][)). In [17], L. Yang extends this result of L. Miller to the case: fl = A+ miE, where E is any compact subset of 8][)) of positive Lebesgue measure that satisfies the Carleson condition

Un} are the intervals that are complementary to E in 8][)). And then (in [16]) J. Thomson and L. Yang obtain this extension of L. Yang in the more general context of the shift on pt(J.l), for 1 < t < oo. The conjecture has recently been established for any measure fl for which there is a nontrivial subarc r of 8][)) such that

i log(~

)dm

> -oo,

with no special assumption made concerning Mlllli; cf. [2]. In [2], the author makes use ofresults in an earlier paper (cf. [1]) that are intricately related to the seminal work of R. Olin and J. Thomson in [12]. Specifically, in [1] the author defines what it means for fl to be strongly inscribed and shows that if fl is such, then indeed dim(M 8 zM) = 1 for each nontrivial member M of Lat(Mz)· To be explicit, fl is said to be strongly inscribed if there is a Jordan subregion W of][)) with rectifiable boundary (we let ww denote harmonic measure on 8W for evaluation at some point in W) with the properties: i) ww(8][]l) > 0, and ii) there is a nonnegative function h in L 00 (ww) such that log( h) E L 1 (ww) and law

for all polynomials

lfl 2 hdww S:

j lfl dfl 2

f.

Since 8W is rectifiable, (i) is equivalent to: m((8W)n(8][)l)) > 0. And this definition is not truly altered if we drop the requirement that W has rectifiable boundary, because if W were any simply connected subregion of][)) that satisfies (i) and (ii), then we could find a Jordan subregion V of W, where V has rectifiable boundary and V itself satisfies (i) and (ii); cf. [14], Proposition 6.23. It is still an open question as to whether or not our general assumptions concerning fl, along with the hypothesis that M(8][)l) > 0, together imply that fl is strongly inscribed. In this brief article we discuss what amounts to the converse of the Fejer-Riesz inequality and find that this converse has close ties to the definition of "strongly inscribed". The Fejer-Riesz inequality, whose statement follows, falls under the general heading of results concerning Carleson measures. For a proof, see [6], Theorem 3.13.


THE FEJER-RIESZ INEQUALITY AND THE INDEX OF THE SHIFT

17

THEOREM 1 (Fejer-Riesz Inequality). Iff E HP (IT)) (0 < p < oo), then [

for 0 :::; 'P

1 1

~

lf(tei'P)IPdt:::;

< 2n. The constant

1 2

7r

lf(ei 11 )1PdB

~ is best possible.

With z fixed in IT), ( r-+ Pz(() := 1 ~:: :~zJ~ 2 is the Poisson kernel on 8IT) for evaluation at z. It is well-known that 8 1fJJ Pz(()dm(() = 1 independent of z, and that if hE L 1 (m), then (by Fatou's Theorem)

J

r

lalfJJ

Pz(()h(()dm(()------; h(O

as z nontangentially approaches ~ for m-a.a. ~ in 81J). One may consult [6] and [8] as good references for these results. We begin with a rather straightforward observation whose proof appears in [4]; see the proof of Lemma 3.1 in this reference. LEMMA 2. LetT/ be a finite, positive Borel measure with support in

ry(81J)) = 0. Then

lim

1-r2

r-->1-

for m-a.a.

~

in 8IT).

111lfJJ

r~wl

2

IT)

such that

dry(w) = 0

Our next result can be viewed as the converse of the Fejer-Riesz inequality. THEOREM 3. Let v be a finite, positive Borel measure with support in IT) such that vlalfJJ « m and IT)= abpe(P 2 (v)). For c > 0, let Bv(c) be the set of all~ in 8IT) such that

1 1

IJ(t01 2dt:::; c ·

J

lfl 2 dv

for all polynomials f. Then Bv (c) is a closed subset of 8IT) and ;:;, 2: ~ (a. e. m) on Bv(c). Furthermore, if Eisa Lebesgue measurable subset of Bv(c) and m(E) > 0, then XE tf_ P 2 (v).

Proof. That Bv(c) is closed is an immediate consequence of the fact that any polynomial is uniformly continuous on JD. Now if 0 < r < 1 and I~ I = 1, then

vT==T2

g(w) := ----=1- r~w

is analytic in a region containing lD and so, by Runge's Theorem, is the uniform limit (on JD) of polynomials. Therefore we can apply our hypothesis to get that

11

which yields: l+r=

1 1

0

(

lg(t012dt:::; c.

1- r 2 ) 2 dt:::;c· 1 - rt

J JII -

l912dv,

1- r 2 r~wl

2

dv(w),

for 0 < r < 1 and any ~ in B 11 (c). Letting r --+ 1 and applying Lemma 2, we find that ;:;_, 2: ~ (a.e. m) on Bv(c). To finish the proof of this theorem, let E be a Lebesgue measurable subset of Bv(c) such that m(E) > 0, and suppose


JOHN R. AKEROYD

18

that XE E P 2 (v); we look for a contradiction. Now h := (1- XE) E P 2 (v) (since XE E P 2 (v)), and in fact, hg E P 2 (v) whenever 0 < r < 1 and~ E E. Arguing as before, with hg now in the place of g, we obtain: ~ ::; h · j~ = 0 a.e. m on E clearly a contradiction. D

It turns out that if f-L is strongly inscribed, then in fact there exists c > 0 such that m(Bil(c)) > 0. En route to this result (Theorem 5, below), we make the following observation. PROPOSITION 4. Let f-L be a finite, positive Borel measure with support in such that P 2(!-L) is irreducible. Then the following are equivalent.

][)I

1) f-L is strongly inscribed. 2) There is a Jordan subregion V ofliJi, where 8V is rectifiable and m((8V) n (8][)1)) > 0, and there is a positive constant M, such that lav lfl2dwv ::; M.

j lfl2df-L

for all polynomials f. Proof. We first assume (1), and so by definition there is a Jordan subregion W of][]) and a nonnegative function h in L 00 (ww) that satisfy certain requirements. One of these requirements, namely that log( h) E L 1 ( ww), guarantees the existence of a bounded analytic function g in W such that g o cp is an outer function (cp is a conformal mapping from lDl onto W) and lgl has "boundary values" equal to h (a.e. ww ). Applying Proposition 2.2 of [1], we can find a Jordan subregion V of W, where V has rectifiable boundary, wv(8lDl) > 0 and lgl 2 r:: > 0 on V. So by the subharmonicity of IJI 2 Igl in W, (2) holds, with M := ~· That (2) implies (1) is immediate, and our proof is complete.D THEOREM 5. Let f-L be a finite, positive Borel measure with support in ID such that P 2 (!-L) is irreducible. If f-L is strongly inscribed, then there exists c > 0 such that m(BJl(c)) > 0.

Proof. Assuming that f-L is strongly inscribed, Proposition 4 provides a Jordan subregion V of ][)I with the properties listed in (2). Let cp be a conformal mapping from ][)I one-to-one and onto V, and let 'ljJ = cp- 1 . Since wv(8liJi) > 0, we can find (cf. [14], Theorem 6.8 and Theorem 3.7) a closed subset E of (8V) n (8lDl), where each point in E is a point of tangency of 8V with 8liJi, such that:

i) m(E) > 0,

ii) for any~ in E and any Stolz angle~ whose closure is contained in VU{O, there is a constant M > 1 such that

iii)

for all z in ~' and if~ E E and r is a smooth arc in V U {0 having nontangential approach in V to ~' then 'lf;(r) is smooth and has nontangential approach in ][)I to 'lf;(~).

Now choose ~ in E. Since ~ is a point of tangency of 8V with 8liJi, there exists s, 0 < s < 1, such that t~ E V whenever s ::; t < 1. Let r be the smooth curve in


THE FEJER-RIESZ INEQUALITY AND THE INDEX OF THE SHIFT ][))U{?/'(~)}

constants

19

defined by "!(t) =?/'(tO, s::::; t::::; 1. Then, by (i)- (iii), there are positive (k = 1, 2, 3) independent of to in [s, 1) such that

ck

length("!( [t 0 , 1]))

ft~ 17/''(t~)ldt

1- I"Y(to)l

1 - 17/J(to~)

I

~~-to~

0 for some integer n, which completes the proof.D QUESTION 6. Does the converse of Theorem 5 hold? That is, if JL is a finite, positive Borel measure with support in lTh such that P 2 (JL) is irreducible, and if m(BM(c)) > 0 for some positive constant c, then is JL strongly inscribed? Indeed, can we even assert that dim(M 8 zM) = 1 for each nontrivial, closed invariant subspace M for the shift on P 2 (JL)?

We conclude this article with a rather anemic response to Question 6 that supports an affirmative answer.


20

JOHN

R.

AKEROYD

REMARK 7. There are other more general forms of the Fejer-Riesz inequality, where the integral on the left is taken over chords of the unit circle and not just over diameters. A converse to this Fejer-Riesz inequality (for chords) can be established, and involves integrals over segments that have nontangential approach in [}) to certain points in 8IDl. Thus, analogues of Bl-'(c) can be defined, where the integral on the left is taken over segments in various Stolz angles. All of this leads to a counterpart of Theorem 5, whose converse appears to be manageable. To this author it seems most likely that if m(BI-'(c)) > 0, then there is a sizeable subset E of Bl-'(c) that is contained in these collections that are analogous to Bl-'(c), and thus the converse of Theorem 5 is likely a consequence of its counterpart in the context of the Fejer-Riesz inequality for chords. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

J. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J., 50 (2001), 705-718. J. Akeroyd, A note concerning the index of the shift, Proc. Amer. Math. Soc., Vol. 130, No. 11 (2002), 3349-3354. C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. Functional Analysis, 63 (1985), 369-404. A. Aleman, S. Richter, C. Sundberg, The majorization function and the index of invariant subspaces in the Bergman spaces, J. Analyse Math., 86 (2002), 139-182. J. B. Conway, L. Yang, Some open problems in the theory of subnormal operators, Holomorphic spaces, Cambridge University Press, 33 (1998), 201-209. P. L. Duren, Theory of HP Spaces, Academic Press, New York, 1970. H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 13-30. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. S. Hruscev, The problem of simultaneous approximation and removal of singularities of Cauchy -type integrals, Trudy Mat. Inst. Steklov 130 (1978), 124-195; English trans!., Proc. Steklov Inst. Math. 130 (1979), no. 4, 133-203. T. L. Kriete, B. D. MacCluer, Mean-square approximation by polynomials on the unit disk, Trans. Amer. Math. Soc., vol. 322, no. 1 (1990), 1-34. T. L. Miller, Some subnormal operators not in A2, J. Functional Analysis, 82 (1989), 296-302. R. F. Olin, J. E. Thomson, Some index theorems for subnormal operators, J. Operator Theory, 3 (1980), 115-142. R. F. Olin, L. Yang, A subnormal operator and its dual, Canad. J. Math., 48 (1996), 381-396. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, BerlinHeidelberg, 1992. J. E. Thomson, Approximation in the mean by polynomials, Ann. Math., 133 (1991), 477-507. J. E. Thomson, L. Yang, Invariant subspaces with the codimension one property in Lt(J.L), Indiana Univ. Math. J., vol. 44, no. 4 (1995), 1163-1173. L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in A1\A2, Mich. Math. J., 42 (1995), 301-310. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE, ARKANSAS

E-mail address: j akeroydCDcomp. uark. edu


72701


A Cauchy-Green Formula on the Unit Sphere in C 2 John T. Anderson and John Wermer ABSTRACT. In 1977 G. Henkin introduced an integral formula for solving Bbf = Jl where J.t is a measure, on the boundary of a smooth strictly convex domain. This result is closely related to a "Cauchy-Green" formula on the sphere (see Chen and Shaw [3]). We give a direct elementary proof of the Cauchy-Green Theorem on the unit sphere and derive Henkin's solution of the equation from this. We also give an application to an approximation result.

ab

1. Introduction

Let n be a domain in the plane, with smooth boundary r. Cauchy-Green formula states that for any ¢ E C 1 (0) and z E n, (1.1)

The classical

¢(z) = _1 [ ¢(() d(- _1 [ a~ d( 1\ d( 27ri lr (- z 27ri } 0 8( ( - z

Note that the first term on the right of (1.1) is a holomorphic function of z in the domain n. In fact, extends continuously to 0, and hence defines an element of the algebra A(O) consisting of functions holomorphic inn and continuous on Of course, if¢ E A(O), (1.1) reduces to the Cauchy integral formula and = ¢. The representation (1.1) has many applications in complex analysis. In the theory of approximation of continuous functions on a compact set K C C by rational functions with poles off K, one is led by considerations of duality to examine measures supported on K. The Cauchy transform of such a measure J.L is defined by

n.

P,(z)

(1.2)

= [ dJ.L(() jK (- Z

The integral defining {1 converges absolutely for almost all z E C. Using (1.1), one can easily show that for any smooth compactly supported function ¢, (1.3)

JK[ 0, n = 0, 1, 2, ... ,p = 1, 2, .... 00 ( -1- ) ~ liml 1 ( -1- ) as topological If A(ap ' n) is nuclear, then A"'(ap ' n) ~ liml ----+ ap,n ---+ ap,n p

p

vector spaces. 1 ( -1- ) is an algebra if A(ap n) satisfies: Here we prove that liml ---+ ap,n ' p

(*) for each p there exist q > p and Mp such that ap,n ap,m :::; Mpaq,n+m (or 1- : : : ; Mp-1- - 1 - ) for all n,m. equivalentely -aq,n+m ap,n ap,rn And then the convolution product is jointly continuous. We also prove that if

(ap,n) does not satisfy(*), then liml 1 ( -1- ) is not an algebra. Therefore liml 1 ( -1- ) ---+

ap,n

p

---+

ap,n

p

is an algebra if, and only if, it is a topological algebra. Thus, if A(ap,n) is nuclear, then A"'(ap,n) is an algebra if, and only if, it satisfies (*). Therefore A(ap,n) and A"'(ap,n) are topological algebras under the normal topology. This is a generalization of the properties of (A, B) and (c-, c-"'). We also study some other important examples of topological algebras with such properties.

2. Definitions and Notation We recall some relevant definitions. Through this section we assume that X is a commutative complex topological algebra with unit element. X is called a locally convex algebra if it is also a locally convex space. In this case its topology can be given by means of a family (11-lla)aE~ of seminorms such that for each index a E ~' there is an index f3 E ~ such that (2.1) for all x, y E X. If relation (2.1) can be replaced by (2.2) for all x, y E X, then we say that X is locally multiplicatively convex (shortly m-convex) algebra. X is called a B 0 -algebra if it is a complete metrizable locally convex algebra. In this case its topology can be given by means of a sequence (11-lln)~=l of seminorms satisfying for n = 1, 2, ... and for all x, y EX. Let (a 1,k), '/ E r, k = 0, 1, ... , be an infinite matrix of positive real numbers. Assume that for each 'f E r there is a 1' E r such that

(2.3) for all k, l = 0, 1, ....


34

HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS

The matrix algebra A(a-y,k) associated with the matrix (a-y,k) is the algebra of all formal complex power series x =

00

E

k=O

XkZk such that, for each 1 E r,

= L a-y,k lxkl < oo. 00

llxii'"Y

k=O

By (2.3), A(a-y,k) is a complete locally convex algebra under the usual linear operations and the convolution product. If r = N, then A(a-y,k) is a B 0 -algebra. If 1 = 1' in (2.3), then A(a-y,k) is an m-convex algebra. We shall be mainly interested in a B 0 matrix algebra A(ap,n)· We note that

L lznYnl < 00

A""(ap,n) = {(zn)~=O

I

V Y = (Yn)~=O

00,

n=O

in A"(ap,n)}.

We have the following: REMARK PROOF.

2.1. A""(ap,n) = A(ap,n)· For each p, the row (ap,n) is an element of A"(ap,n)· Therefore, if 00

E

z E A""(ap,n), then

n=O

lznap,nl < oo and hence z E A(ap,n)·

D

3. A Matrix Algebra and its a-dual 3.1.

PROPOSITION

A"(ap n) '

= liml ----t

= {(yn)~=O

00 ( -1- ) ap,n

p

PROOF.

=

M

Let y

-1,

00

x

E

E

1 00 ( - 1- ) ; liml then sup IYn -ap,n -;: ap,n n

E

1 sup IYnn ap,n

-1 -1

----7

p

1 then there exists an increasing sequence (np) such that IYn P -ap,np

define the sequence x

=

as follows: Xn

(xn)~=O

We claim that x E A(ap,n): For arbitrary q, llxllq

=

+ """"' L....

1 for Nevertheless n=O lxnYnl = n=O ap,np p P ap,np p P every n. This is a contradiction to the assumption that y E A"(ap,n)· D

3.2. The inductive limit topology is stronger than the normal topology in A"(ap,n)· PROPOSITION


ON

35

a-DUAL ALGEBRAS

00 ( -1- ) PROOF. Let us recall that 11.11 is a continuous seminorm m liml ---+ ap,n

if,

and

~

IIYII

only

if,

Cp IIYIIzoo =

for

Cps~p

each p,

n~O

exists

IYna:.n I for every y E l 00

Let y E l 00 (a:.J, then for each n.

there

s~p

IYna:.n I = M
panda sequence U = (un)~=O n = 0, 1, ... PROPOSITION 3.3. If A(ap,n) is nuclear, then A"(ap,n), endowed with the nor1-). 00 ( -1 1(-ap,n - ) ~ liml mal topology, is such that A"(ap ' n) ~ liml ---+ ap,n ---+ p

p

PROOF. Due to the nuclearity of A(ap,n) it is easy to see that 1-) ~ liml 1 00 ( -1 liml (-ap,n -). By the previous Proposition we know that the inductive ---+ ---+ ap,n p

p

limit topology is stronger than the normal topology in A"(ap,n)· 1 ( -1- ) . Let (en)~=l Let 11·11 be a seminorm in liml be the sequence of the --+ ap,n p

canonical vectors and note that (en)~=l

E 11 (ap,n -1- ) for each p.

exists Cp > 0 such that llenll ~ Cvllenllzoo (xn)~=l = (llenll)~=l· We will prove that (xn)~=l Indeed, llxnllv 00

Cq

L

n=l

Un
-.il ~

iE/

1}. It is a straightfor-

ward matter to check that if V and W are two subsets of an algebra, then the property (x, y E V =? xy E W) implies the property (x, y E f(V) =? xy E f(W)). PROPOSITION 3.5. If the matrix algebra A(ap,n) satisfies (*), then 1~l p

is a topological algebra under the convolution product.


1

(a:.J

36

HUGO ARIZMENDI, ANGEL CARRILLO, AND LOURDES PALACIOS

Let x, y E l 1(-1-). From condition(*), let q

PROOF.

aq1,n:::; MPa:,k

f £:

ap,~-k· The~Pii:YIIq

=

f ( £:

n~ 0 l(xy)nl

a01,n

n~O~k~OIXkYn-kll

=

C~o

C~o

1,n:::;

a0

I) =

1 n=O k=O IXkYn-kl ,-!-:::; q,n n=O k=O Mp lxk;f;IIYn-ka p, p,n _k

Mp

> p and Mp be such that

IYnl a:,n) = Mp llxllp IIYIIP < 00. 1 1(--) is an algebra. From the previous proof it is easy to see that So far, liml --+ ap,n lxnl a:,n)

.J'

p

if X E l 1 ( a:.J and y E l 1 ( a.1 then xy E l 1 ( a:.J and llxyllq :::; MpllxllriiYIIs, where q and Mp are any two numbers that satisfy (*) for p = max(r, s). 1(-1-); then Now let 0 be a neighborhood of the origin in liml --+ ap,n p

n= r

Cg1 V(O, II-IlP '€p))' where €p

> 0 for all p

~ 1.

For each p ~ 1, let qp and Mp be two numbers that satisfy (*) for p and let us take

0< 8p 0 and an uncountable set £ c Ker 0 such that r(A) + E for all a E A. By definition of r(A), there exists y E X such that supaEA IIY- all < r(A) +E. Clearly, y =:;A x. (b). By (a), if x tf. A+ r(A)B(X), there exists y E X such that y ::;A x, and hence, f(y) ::; f(x). Thus, the infimum off over X equals the infimum over A+ r(A)B(X). Moreover, since X is a dual space and f is w*-lsc, it attains its minimum over any w*-compact set. Thus f actually attains its minimum over X as well. Since the norm on X is w*-lsc, so is ¢A,p for every p. (c). Consider {x EX: x ::;A xo}. Let {xi} be a totally ordered subset. Let z be a w*-limit point of Xi. Since the norm is w*-lsc, we have PROOF.

llx- all >

liz- all ::; liminf llxi- all

=

inf llxi-

all

for all a EA.

Thus the family {xi} is ::;A-bounded below by z. By Zorn's lemma, there is a x 1 E mx(A) such that x 1 ::;A x 0 . Now let x 0 be a minimum for f. There is a x 1 E mx(A) such that x 1 Clearly, f attains its minimum also at x 1 .

::;A

x0 . D

2.3. (a) It follows that for any bounded set A, minA C A+ r(A)B(X). This improves the estimates in [9] or [18]. (b) Apart from ¢A,p, there are many examples of A-monotone and w*lsc f : X = Z* ~ Jlt+. One particular example that has been treated extensively in [4] is the function ¢1-' defined by ¢1-'(x) = JA llx- all 2df.L(a), where /L is a probability measure on a compact set A 0, there exists y E Y such that

(1)

llY- all ::; llx- all + E forall a E A. (b) We say that Y is an A-C-subspace of X if we can take E = 0 in (a). (c) If A is a family of subsets of X, we say that X has the (almost) A- IP if X is an (almost) A-C-subspace of X**.


BANDYOPADHYAY AND DUTTA

46

Some of the special families that we would like to give names to are : (i) :F = the family of all finite sets, (ii) K = the family of all compact sets, (iii) B = the family of all bounded sets, (iv) P =the power set. Since these families depend on the space in which they are considered, we will use the notation :F(X) etc. whenever there is a scope of confusion. REMARK 2.5. (a) Note that :F-C-subspaces were called central (C) subspaces in [3], P-C-subspaces were called almost constrained (AC) subspaces in [1, 2]. Also if X has the :F-IP, it was said to belong to the class (GC) in [18, 3], and the P-IP was called the Finite Infinite Intersection Property (IPJ,oo) in [7, 2]. (b) The definition of almost A-C-subspace is adapted from the definition of almost central subspace defined in [17]. The exact analogue of the definition in [17] would have, in place of condition (1), sup IIY- all

aEA

:=:; sup llx- all +E. aEA

Clearly, our condition is stronger. We observe below (see Proposition 2.7) that this definition is more natural in our context. (c) By the Principle of Local Reflexivity (henceforth, PLR), any Banach space has the almost :F-IP. More generally, if Y is a!l ideal in X (see definition below), then Y is an almost :F-C-subspace of X. DEFINITION 2.6. A subspace Y of a Banach space X is said to be an ideal in X if there is a norm 1 projection P on X* with ker( P) = Y .L. PROPOSITION 2. 7. Let Y be a subspace of a Banach space X. Let A be a family of bounded subsets of Y. Then the following are equivalent : (a) Y is an almost A-C-subspace of X (b) for all A E A and p : A----+ JR+, if naEABx [a, p(a)] -1- 0, then for every E > 0, naEABy[a,p(a) +c]-:f- 0. (c) for every bounded p, the infimum of ¢A,p over X andY are equal. PROOF. Equivalence of (a) and (b) is immediate and does not need A to be bounded. (a) ==> (c). Let Y be an almost A-C-subspace of X, A E A and p : A ----+ JR+ be bounded. Let M = supp(A). Let E > 0. By definition, for x EX, there exists y E Y such that

IIY -all :=:; llx - all + E for all a E A.

It follows that p(a)IIY-

all:=:; p(a)llx- all+ p(a)c :=:; p(a)llx- all+ Me for all a EA.

and hence, ¢A,p(Y)

:=:; ¢A,p(x) +Me.

Therefore, inf ¢A,p(Y):::; inf ¢A,p(X) As

E

+Me.

is arbitrary, the infimum of ¢A,p over X andY are equal.


CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS

47

(c) ::::} (a). Let A E A, x E X and c > 0. We need to show that there exists y E Y such that IIY - all ::=; llx - all + c for all a EA. If x E Y, nothing to prove. Let x E X\ Y. Let N = supaEA llx- all· Let p(a) = 1/llx- all· Since x tf. Y and A ~ Y, p is bounded. Then ¢A,p(x) = 1, and therefore, inf ¢A,p(X) ::=; 1. By assumption, inf ¢A,p(Y) = inf ¢A,p(X) ::=; 1, and so, there exists y E Y, such that ¢A,p(y) ::=; 1+ c/N. This implies IIY- all ::=; llx- all+ cllx- aii/N ::=; llx- all+ c for all a EA. D As noted before, by PLR, any Banach space has the almost F-IP. And therefore, the result of [18] follows. PROPOSITION 2.8. Let A and A 1 be two families of subsets of Y such that for every A E A and c > 0, there exists A1 E A1 such that A~ A1 + cB(Y). If Y is an almost A 1 -C-subspace of X, then Y is an almost A-C-subspace of X as well. Consequently, any ideal is an almost K-C -subspace and any Banach space has the almost K-IP. In particular, if A is a compact subset of X and p: A -----> ~+ is bounded, then the infimum of¢ A,p over X and X** are the same. PROOF. Let A E A and c > 0. By hypothesis, there exist A1 E A 1such that A ~ A 1 + cB(Y). Let x E X. Since Y is an almost A 1 -C-subspace of X, there exists y E Y such that

IIY- a1ll

llx- a1ll + c/3 for all a1 EA1. Now fix a EA. Then there exists a1 E A1 such that lla- a1ll < c/3. Then IIY- all < IIY- a1ll + lla- a1ll ::=; llx- a1ll + 2c/3 < llx- all + II a- ad + 2c/3 ::=; llx- all+ c. ::=;

Therefore, Y is an almost A-C-subspace of X as well. Since any Banach space has the almost F-IP, by the above, it has the almost K-IP too. The rest of the result follows from Proposition 2. 7. D EXAMPLE 2.9. Vesely [18] has shown that if A is infinite, the infimum of ¢A,p over X and X** may not be the same. His example is X = c0 , A = {en : n ;::: 1} is the canonical unit vector basis of co and p = 1. Then inf ¢A,p(X) = 1 and inf ¢A,p(X**) = 1/2. The example clearly also excludes countable, bounded, or, taking AU {0}, even weakly compact sets. Thus c0 fails the almost B-IP, almost P-IP and if A is the family of countable or weakly compact sets, then c0 fails the almost A-lP too. Stronger conclusions are possible for A-lP. LEMMA 2.10. Let Y be a subspace of a Banach space X. For A ~ Y, the following are equivalent : (a) For every A-monotone f : A -----> ~+ and x E X, there exists y E Y such that f(y) ::=; f(x). (b) For every p : A -----> ~+ and x E X, there exists y E Y such that ¢A,p(y):::; ¢A,p(x). (c) For every continuous p : A -----> ~+ and x E X, there exists y E Y such that ¢A,p(Y):::; ¢A,p(x). (d) For every bounded p : A -----> ~+ and x E X, there exists y E Y such that ¢A,p(y) ::=; ¢A,p(x).


48


(e) Any family of closed balls centred at points of A that intersects in X also intersects in Y. (f) for any x EX, there exists y E Y such that y ::;Ax. It follows that whenever any of the above conditions is satisfied, for every Amonotone f : A ------> ~+, the infimum of f over X and Y are equal and if A has a weighted Chebyshev centre in X, it has a weighted Chebyshev centre in Y.

PROOF. (a)=? (b)=? (c), (b)=? (d) and (e)? (f)=? (a) are obvious. (c) or (d) =? (!). As in the proof of Proposition 2. 7, let p(a) = 1/llx- all· Then pis continuous and bounded and ¢A,p(x) = 1. Thus, there exists y E Y such that rPA,p(Y)::::; 1. This implies IIY- all ::::; llx- all for all a EA. 0 We now conclude the discussion so far by obtaining the extension of Theorem 1.2. THEOREM 2.11. For a Banach space X and a family A of bounded subsets of X, the following are equivalent : (a) X has the A-IF. (b) For every A E A and every f : X** ------> ~+ that is A -monotone and w*-lsc, the infimum off over X** and X are equal and is attained at a point of X. (c) For every A E A and every p, the infimum of ¢A,p over X** and X are equal and is attained at a point of X. Moreover, the point in (b) or (c) can be chosen to be ::;A -minimal. We now study different aspects of A-C-subspaces. DEFINITION 2.12. Let Y be a subspace of a Banach space X. Let A x EX and x* E B(X*), define U(x, A, x*) L(x,A,x*)

=

inf{x*(y)

+ llx- Yll

c::;;;

Y. For

:yEA}

sup{x*(y)- llx- Yll :yEA}

The following lemma is in [1]. We include the proof for completeness. LEMMA 2.13. Let Y be a subspace of a Banach space X and A c::;;; Y. For x1,x2 EX, Xz ::;A x1 if and only ifforallx* E B(X*), U(xz,A,x*)::::; U(x1,A,x*). PROOF. If Xz ::;A X1, then for all x* E B(X*), x*(y) + llxz- Yll ::::; x*(y) + llx1- Yll· And therefore, U(xz,A,x*)::::; U(x1,A,x*). Conversely, suppose llxz - Yoll > llx1 - Yoll for some Yo E A. Then there exists c: > 0 such that llxz -Yo II - c: :::0: llx1 - Yo II· Choose x* E B(X*) such that llx1- Yo II ::::; llxz- Yo II- c: < x*(xz- Yo)- c:/2. Thus U(x1, A, x*)::::; x*(yo) + llx1Yo II < x*(xz)- c:/2 < U(xz, A, x*). 0 REMARK 2.14. Instead of B(X*), it suffices to consider the unit ball of any norming subspace of X*. We compile in the following propositions several interesting facts about A-Csubspaces and the A-IP. PROPOSITION 2.15. Let Y be a subspace of a Banach space X. For a family A of subsets of Y, the following are equivalent : (a) Y is an A-C-subspace of X



49

(b) for every x E X and A E A, there exists y E Y such that U(y, A, x*) ::=; U(x, A, x*) for every x* E B(X*). (c) for any A E A, Ay,x ~ Y. PROOF. This follows from Lemma 2.13 and the definition of Ay,x.

0

COROLLARY 2.16. X has the P-IP if and only if for every x** E X**, there exists x E X such that x is dominated on B(X*) by the upper envelop of x** considered as a function on B(X*) equipped with the w*-topology. PROOF. Observe that for any x EX, U(x,X, ·) = x on B(X*) and for x** E X**, U(x**, X, x*) is the upper envelop of x** considered as a function on B(X*) equipped with the w*-topology (see [8]). 0 PROPOSITION 2.17. (a) Let X be a Banach space and let Y be a subspace of X. Let A be a family of subsets of Y and let A 1 be a subfamily of A. IfY isaA-C-subspace of X, then Y is a A1-C-subspace of X as well. In particular, P-IP implies B-IP implies K-IP implies F-IP. (b) 1-complemented subspaces are A-C-subspaces for any A. (c) Let Z ~ Y ~ X and let A be a family of subsets of Z. If Z is an A-C-subspace of X, then Z is an A-C-subspace of Y. And, if Y is an A-C -subspace of X, then the converse also holds. PROOF. The proof follows the same line of argument as in [3, Proposition 2.2]. We omit the details. 0 PROPOSITION 2.18. For a family A of subsets of a Banach space X, the following are equivalent : (a) X has the A-lP (b) X isaA-C-subspace of some dual space. (c) for all A E A and p : A-+ JR+, nf= 1Bx [ai, p(ai) + c] =/:- 0 for all finite subset {a 1, a2, ... , an}~ A and for all c > 0 implies naEABx[a, p(a)] =f. 0. In particular, any dual space has the A-lP for any A. Let S be any of the families F, K, B or P. The S-IP is inherited by S-C-subspaces, in particular, by 1-complemented subspaces. PROOF. Clearly, (a)::::} (b), while (c)::::} (a) follows from the PLR. (b)::::} (c). Let X be an A-C-subspace of Z*. Consider the family {Bz· [a, p(a)+ c] : a E A, c > 0} in Z*. Then, by the hypothesis, any finite subfamily intersects. Hence, by w*-compactness, naEABz· [a, p(a)] =1- 0. Since X is an A-C-subspace of Z*' we have naEABx [a, p(a)] =1- 0. 0 The following result significantly improves [3, Proposition 2.8] and provides yet another characterization of the A-lP. PROPOSITION 2.19. Let Y be an almost F-C subspace of a Banach space X. Let A be a family of subsets of Y. If Y has the A-lP, then Y is an A-C-subspace of X. In particular, the conclusion holds when Y is an ideal in X. PROOF. Let x EX, A EA. Since Y be an almost F-C subspace of X, for all finite subset {a1,a2, ... ,an}~ A and for all c > 0, nf= 1By[ai, llx- ail!+ c] =/:- 0. Since Y has the A-lP, by Proposition 2.18(c), naEABy[a, llx- all] =f. 0. 0 Since X is always an ideal in X**, the following corollary is immediate.


50


COROLLARY 2.20. For a Banach space X and a family A of subsets of X, the following are equivalent : (a) X has A-lP. (b) X is an A -C -subspace of every superspace Z in which X embeds as an almost :F -C subspace. (c) X is an A -C -subspace of every superspace Z in which X embeds as an ideal.

3. Strict convexity and minimal points PROPOSITION 3.1. If a Banach space X is strictly convex, then for every A X, min A= mx(A).

~

PROOF. As we have already observed, min A~ mx(A). Let x 0 E mx(A) and x 0 tJ_ minA. Then there is an x E X such that x-/= xo and x :::;A xa. Since xo E mx(A), we must have llx- all = llxo- all for all a E A. Since X is strictly convex, ll(x + xo)/2- all < llxo- all for all a. This contradicts that x 0 E mx(A). Hence xo EminA. D REMARK 3.2. If X is strictly convex, by a similar argument, for every x 0 EX, there is at most one x 1 E mx (A) such that x 1 :::;A x 0 . Thus for a strictly convex dual space, for every x 0 E X*, there is a unique xi E mx• (A) such that xi :::;A x 0. PROPOSITION 3.3. Let X be strictly convex. Let A be a compact subset of X. For each continuous p, A admits at most one weighted Chebyshev centre. PROOF. Suppose A admits two distinct weighted Chebyshev centres x 0 , x1 E X. Then ¢A,p(x 0 ) = ¢A,p(xl) = r (say). Then for all a E A, we have x 1, xo E Bx[a, rj p(a)]. By rotundity z = (x 1 + x 0 )/2 is in the interior of Bx[a, rj p(a)] for all a. Thus, p(a)llz- all < r, for all a. Since pis continuous, ¢A,p(z) < r, which contradicts that minimum value is r. D THEOREM 3.4. Let X be a Banach space such that ( i) X has the :F- IP; and (ii) for every compact set A~ X, mx(A) is weakly compact. Then X has the K-IP. Moreover, if X** is strictly convex, then the converse also holds. PROOF. Let X have the :F-IP and for every compact set A ~ X, let mx(A) be weakly compact. Observe that for any B ~A, we have mx(B) ~ mx(A). Let A~ X be compact and let x** EX**. By Lemma 2.10, it suffices to show that there is a zoE X such that llzo- all :::; llx**- all for all a EA. Let {an} be a norm dense sequence in A. Take a sequence ck ----+ 0. By compactness of A, for each k, there is a nk such that A ~ u~k Bx [an, Ckl· Since X has the :F-IP, there exists Zk E n~k Bx [an, llx**- ani I] and Zk E mx ({a1, a2 · · · ank}) ~ mx(A). Then llzk-all:::; llx**-all+2ck for all a EA. Now, by weak compactness of mx(A), we have, by passing to a subsequence if necessary, Zk----+ z 0 weakly for some z 0 EX. Since the norm is weakly lsc, we have llzo-all:::; liminf llzk-all:::; llx**-all for all a EA. Conversely, let X have the K-IP and X** be strictly convex. Let A ~ X be compact. It is enough to show that any sequence {xn} ~ mx(A) has a weakly convergent subsequence. Without loss of generality, we may assume that {xn} are



51

all distinct. By Remark 2.3 (a), mx(A) ~ A+ r(A)B(X) is bounded. Let x** be a w*-cluster point of {xn} in X**. It suffices to show that x** EX. Suppose x** E X** \X. Since X has the K-IP, there exists x 0 E mx(A) such that llxo - all ~ llx** - all for all a E A. Since X** is strictly convex, ll(x** +xo)/2-all < llx** -all for all a EA. Since (x** +xo)/2 EX** \X, by K-IP again, there exists zo E mx(A) such that llzo- all ~ ll(x** +xo)/2- all < llx** -all for all a EA. Since A is compact, there exists E: > 0 such that llzo -all < llx** -all - E: for all a E A. Observe that llzo- all

< llx**- all- E:

~ liminf llxn- all- E: for all a EA. n

Therefore, for every a E A, there exists N(a) EN such that for all n ~ N(a), llzo- all < llxn- all - E:. By compactness, there exists N E N such that llzo- all < llxn- all - c/4 for all n ~ N and a E A. Thus, zo ~A Xn for all n ~ N. Since Xn E mx(A) and X is strictly convex, zo = Xn for all n ~ N. This contradiction completes the proof. 0 REMARK 3.5. In proving sufficiency, one only needs that {zk} has a subsequence convergent in a topology in which the norm is lsc. The weakest such topology is the ball topology, bx. So it follows that if X has the F-IP and for every compact set A~ X, mx(A) is bx-compact, then X has the K-IP. Is the converse true? COROLLARY 3.6. [4, Corollary 1] Let X be a reflexive and strictly convex Banach space. Let A~ X be a compact set. Then min(A) is weakly compact. REMARK 3.7. Clearly, our proof is simpler than the original proof of [4]. If Z is a non-reflexive Banach space with Z*** strictly convex, then X = Z* is a non-reflexive Banach space with K-IP such that X** is strictly convex. Thus, our result is also stronger than [4, Corollary 1].

4. L1-preduals and P1-spaces Our next theorem extends [3, Theorem 7], exhibits a large class of Banach spaces with the K-IP and produces a family of examples where the notions ofFG-subspaces and K-G-subspaces are equivalent. DEFINITION 4.1. (a) [12] A Banach space X is called an L 1 -predual if X* is isometrically isomorphic to L 1 (J.L) for some positive measure J.l· (b) [11] A family {Bx[xi,ri]} of closed balls is said to have the weak intersection property if for all x* E B(X*) the family {BJR[x*(xi), ri]} has nonempty intersection in JR. THEOREM 4.2. For a Banach space X, the following are equivalent : (a) X is a K-G-subspace of every superspace (b) X is a K-G-subspace of every dual superspace (c) X is a F -G -subspace of every superspace (d) X is an almost F -G -subspace of every superspace (e) X is a F-G-subspace of every dual superspace (!J X is an almost F-G-subspace of every dual superspace (g) X is an L 1 -predual.



52

PROOF. Observe that if X ..x + Py)

>..Px + Py, and

llxll = IIPxll + llx- Pxll

for all x, y EX, >..scalar. In [3], it was shown that semi-L summands are F-C-subspaces. Basically the same proof actually shows that PROPOSITION 5.3. A semi-L-summand is an A-C-subspace for any A. Our next result concerns proximinal subspaces. DEFINITION 5.4. A subspace Z of a Banach space X is called proximinal if for every x EX, there exists zo E Z such that llx- zoll = d(x, Z) = infzEZ llx- zll· The map Pz(x) = {zo E Z : llx- zoll = infzEZ llx- zll} is called the metric projection. PROPOSITION 5.5. Let Z ~ Y ~X, Z proximinal in X. (a) Let A be a family of subsets ofY/Z. Let A' be a family of subsets of Y such that for any x E X and A E A, there exists A' E A' such that for any a+ Z E A, {a+ Pz(x- a)} n A' =f. 0. Suppose Y is a A'-C-subspace of X. Then Y/Z isaA-C-subspace of XjZ. Let S be any of the families F, B or P. (b) IfY is a S(Y)-C-subspace of X, then Y/Z is a S(Y/Z)-C-subspace of

x;z.

(c) Suppose the metric projection has a continuous selection. Then, if Y is a K(Y)-C-subspace of X, Y/Z is a K(Y/Z)-C-subspace of X/Z. (d) Let Z ~ Y ~X*, Z w*-closed in X*. If Y is a S(Y)-C-subspace of X*, then Y/Z is a S(Y/Z)-C-subspace of X* /Z, and hence, has the S(Y/Z)-IP. (e) Let X have the S(X)-IP. Let M ~ X be a reflexive subspace. Then X/M has the S(X/M)-IP. PROOF. (a). Let A E A and x + Z E XjZ. Choose A' as above. Then, for a + Z E A, there exists z E Pz (x - a) (depending on x and a) such that a+ z E A'. Since Y is a A'-C-subspace of X, there exists y0 E Y such that IIYo - a - zll ::; llx - a - zll for all a + Z E A. Clearly then IIYo - a+ Zll ::; IIYo- a- zll :S llx- a- zll = llx- a+ Zll·



55

If S is the family under consideration in (b) and (c) above and A = S (Yj Z), then for any choice of A' as above, S(Y) ~ A'. Hence, (b) and (c) follows from (a). For (d), we simply observe that any w*-closed subspace of a dual space is proximinal. And (e) follows from (d) . D

As in [3, Corollary 4.6], we observe PROPOSITION 5.6. Let Z ~ Y ~ X, Z proximinal in Y andY is a semi-Lsummand in X. Then YjZ is a P-C-subspace of XjZ. Let us now consider the c0 or fp sums. THEOREM 5.7. Let r be an index set. For all a E r, let Ya be a subspace of Xa. Let X andY denote resp. the co or fp (1:::; p:::; oo) sum of Xa 'sand Ya 's. (a) For each a E r, let Aa be a family of subsets ofYa such that {0} E Aa and for any A E Aa, there exists B E Aa such that A U { 0} ~ B. Let A be a family of subsets of Y such that for any a E r, the asection of any A E A belongs to Aa. Then Y is an A-C-subspace of X if and only if for each a E r, Ya is an Au-G-subspace of Xa. LetS be any of the families :F, K, B or P. (b) Y is a S(Y)-C-subspace of X if and only if for any a E r, Ya is a S(Ya)-C-subspace of Xa. (c) The S-IP is stable under lp-sums (1:::; p:::; oo). PROOF. (a). The proof is very similar that of to [3, Theorem 4.7]. We omit the details. (c). Xa has S-IP if and only if Xa is a S-O-subspace of some dual spaceY;. Now the fp-sum (1:::; p:::; oo) of Y;'s is a dual space. D REMARK 5.8. The result for :F-IP has already been noted by [18] with a much different proof. The stability of the P-IP under £1 -sums is noted in [15] again with a different proof. [18] also notes that :F-IP is stable under c0 -sum. And Corollary 4.3 shows that c0 has the K-IP. However, we do not know if the K-IP is stable under cosums. As for the B-IP or P-IP, we now show that co-sum of any infinite family of Banach spaces lacks the B-IP, and therefore, also the P-IP. This is quite similar to Example 2.9. PROPOSITION 5.9. Let r be an infinite index set. For any family of Banach spaces Xa, a E r, X = EBc 0 Xa lacks the B-IP. PROOF. For each a E f, let Xa be an unit vector in Xa and define ea EX by if (3 =a otherwise Then the set A= {ea: a E f} is bounded and the balls Bx··[ea, 1/2] intersect at the point (1/2xa) E X**, but the balls Bx [ea, 1/2] cannot intersect in X. D REMARK 5.10. As before, taking AU{O}, it follows that X lacks the A-IP even for A = weakly compact sets. Coming to function spaces, we note the following general result.


56


PROPOSITION 5.11. Let Y be a subspace of a Banach space X and A be a family of subsets of Y. (a) For any topological space T, ifC(T, Y) isaA-C-subspace ofC(T,X), then Y is a A-C -subspace of X. Moreover, if C(T, X) has A-lP, X has A-lP. (b) Let (0, I:, JL) be a probability space. If for some 1 ::; p < oo, LP(JL, Y) is a A-C-subspace of LP(JL, X), then Y is a A-C-subspace of X. Moreover, if LP(JL, X), has A-IP, then X has A-lP. PROOF. For (a) and (b), let F(X) denote the corresponding space offunctions and identify X with the constant functions. In (a), point evaluation and in (b), integral over 0 gives us a norm 1 projection from F(X) onto X. Thus X inherits A-IP from F(X). Now suppose F(Y) is an A-C-subspace of F(X). Let P : F(Y) -+ Y be the above norm 1 projection. Let x E X and A E A. Then, there exists g E F(Y) such that llg-all::; llx-all for all a EA. Let y = Pg. Then, lly-all::; llg-all::; llx-all for all a EA. 0 The following Proposition was proved in [3]. PROPOSITION 5.12. (a) Let X has Radon Nikodym Property and is 1cornplemented in Z* for some Banach space Z. Then for 1 < p < oo, LP(JL,X) is 1-complemented in Lq(JL, Y)* (1/p+ 1/q = 1), and hence has the P-IP. (b) Suppose X is separable and 1-complemented in X** by a projection P that is w*-w universally measurable. Then for 1 ::; p < oo LP (JL, X) is 1-complemented in Lq(JL,X*)* (1/p+ 1/q = 1), and hence has the P-IP. Since the B-IP or P-IP is inherited by 1-complemented subspaces and c0 lacks the B-IP, the next result follows essentially from the arguments of [17]. PROPOSITION 5.13. (a) Let X be a Banach space containing c0 and let Y be any infinite dimensional Banach space. Then X ®c: Y fails the B-IP and P-IP. (b) If C(K, X) has the B-IP, then either K is finite or X is finite dimensional. C(K,X) has the P-IP if and only if either (i) K is finite and X has the P-IP or (ii) X is finite dimensional and K is extremally disconnected. (c) For any nonatomic measure space (O,I:,JL) and a Banach space X containing c0 , L 1 (JL, X) fails the B-IP. In the next Proposition, we prove a partial converse of Proposition 5.11 (a) when Y is finite dimensional and K is compact and extremally disconnected. PROPOSITION 5.14. LetS be any of the families :F, K, B or P. Let Y be a finite dimensional a S (Y) -C -subspace of a Banach space X. Then for any extremally disconnected compact space K, C(K, Y) is a S(C(K, Y))-C-subspace ofC(K,X). PROOF. We argue similar to the proof of [3, Proposition 4.11]. Let K be homeomorphically embedded in the Stone-Cech compactification ;J(f) of a discrete set r and let ¢ : ;J(f) -+ K be a continuous retract. Let A E S(C(K, Y)) and g E C(K, X). Note that since Y is finite dimensional, by the defining property of ;J(f), any Y-valued bounded function on r has a norm preserving extension



57

in C(,B(r), Y). Thus C(,B(f), Y) can be identified with ffieoo(r) Y. Lift A to this space. In view of Theorem 5.7, this space is S(Y)-C-subspace of EB 00 X. This latter space contains C(,B(r), X). Thus by composing the functions with ¢, we get a f E C(K, Y) such that II/- hll ~ 119- hll for all h E A. Hence the result. 0 And now for a partial converse of Proposition 5.11(b). THEOREM 5.15. Let Y be a separable subspace of X. IJY is a P-C-subspace of X, then for any standard Borel space f2 and any f7-jinite measure J.L, Lp(f..L, Y) is a P-C-subspace of Lp(f..L, X). PROOF. Let f E Lp (J.L, X). Since Y is a P-C-subspace of X, for each x E X,

nyEYBy[y, llx- YIIJ #0. Define a multi-valued map F : f2 --. Y, by F(t) = {

nyEY

{f(t)}

By[y,

llt(t)- YIIJ

if

f(t) EX\ Y

if

f(t)EY

Let G = {(t, z) : z E F(t)} be the graph of F. Claim : G is a measurable subset of n X y. To establish the claim, we show that is measurable. Since Y is separable, let {Yn} be a countable dense set in Y. Observe that z tJ. F(t) if and only if either f(t) E Y and z "# f(t) or f(t) E X \ Y and there exists Yn such that liz- Ynll > 11/(t)- Ynll· And hence,

cc

cc = {t(t) E Y

and

z "# f(t)} U U {t(t) EX\ Y

and

liz- Ynll > llf(t)- Ynll}

n~l

is a measurable set. By von Neumann selection theorem, there is a measurable function 9: f2--. Y such that (t,9(t)) E G for almost all t E fl. Observe that ll9(t)11 ~ 11/(t)ll for almost all t. Hence 9 E Lp(f..L, Y). Also for any h E Lp(f..L, Y) we have ll9(t)- h(t)ll ~ llf(t)- h(t)ll for almost all t. Thus, 119- hiiP ~ II/- hiiP for all h E Lp(f..L, Y). 0 QuESTION 5.16. Suppose Y is a separable IC-C-subspace of X. Let (0, E, J.L) be a probability space. Is LP(J.L, Y) a IC-C -subspace of LP(J.L, X)? REMARK 5.17. This question was answered in positive in [3] for F-C-subspaces and we did it for P-C-subspaces. Both the proofs are applications of von Neumann selection Theorem. The problem here is for a compact set A in LP(J.L, Y) and wE f2 the set {!(w) : f E A} need not be compact in Y. ACKNOWLEDGEMENTS. Partially supported by a DST-NSF grant no. RP041/2000. The first-named author availed this grant to visit Southern Illinois University at Edwardsville, USA in May-June 2002 and attended the Fourth Conference on Function Spaces, where he presented a talk based on this work. He would like to thank Professor K. Jarosz for the warm hospitality and a wonderful conference. We also thank the referee for suggestions that improved the paper.


58


References [1] P. Bandyopadhayay, S. Basu, S. Dutta and B. L. Lin Very nonconstrained subspaces of Banach spaces, Preprint 2002. [2] P. Bandyopadhayay and S. Dutta, Almost constrained subspaces of Banach spaces, Preprint 2002. [3] Pradipta Bandyopadhyay and T. S. S. R. K. Rao, Central subspaces of Banach spaces, J. Approx. Theory, 103 (2000), 206-222. [4] B. Beuzamy and B. Maurey, Points minimaux et ensembles optimaux dans les espaces de Banach, J. Functional Analysis, 24 (1977), 107-139. [5] J. Diestel, Geometry of Banach Spaces, selected topics, Lecture notes in Mathematics, Vo1.485, Springer-Verlag (1975). [6] J. Diestel and J. J. Uhl, Jr., Vector measures, Mathematical Surveys, No. 15, Amer. Math. Soc., Providence, R. I. ( 1977). [7] G. Godefroy, Existence and uniqueness of isometric preduals : a survey, Banach space theory (Iowa City, IA, 1987), 131-193, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [8] G. Godefroy and N. J. Kalton, The ball topology and its applications, Banach space theory (Iowa City, IA, 1987), 195-237, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [9] G. Godini On minimal points, Comment. Math. Univ. Carolin., 21 (1980), 407-419. [10] P. Harmand D. Werner and W. Werner, M -ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547, Springer-Verlag, Berlin, 1993. [11] 0. Hustad, Intersection properties of balls in complex Banach spaces whose duals are Lr spaces, Acta Math., 132 (1974), 283-313. [12] H. E. Lacey, Isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. [13] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc., No. 48, 1964. [14] A. Lima, Complex Banach spaces whose duals are L 1 -spaces, Israel J. Math., 24 (1976), 59-72. [15] T. S. S. R. K. Rao, Intersection properties of balls in tensor products of some Banach spaces II, Indian J. Pure Appl. Math., 21 (1990), 275-284. [16] T. S. S. R. K. Rao, On ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001), 595-609. [17] T. S. S. R. K. Rao Chebyshev centers and centrable sets, Proc. Amer. Math. Soc., 130 (2002), 2593-2598. [18] L. Vesely, Generalized centers of finite sets in Banach spaces, Acta Math. Univ. Comen., 66 (1997), 83-115. (Pradipta Bandyopadhyay) STAT-MATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, E-mail : pradipta@isical. ac. in (S Dutta) STAT-MATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, E-mail : sudipta..r@isical. ac. in



The boundary of the unit ball in H 1-type spaces Paul Beneker and Jan Wiegerinck ABSTRACT. This is a survey on the extreme, exposed and strongly exposed

boundary points of the unit ball in various H 1 -type spaces.

1. Introduction

Theorems like the Krein-Milman theorem and Phelps' theorem assert the existence of many extreme points in certain convex sets. However, to decide whether a particular point in the boundary of a convex set is extreme, exposed or strongly exposed is quite a different question. In this paper we survey what is known in the very special situation of the unit ball of certain Hardy spaces. Of course we refer to the literature for the majority of the proofs. But we have included some, and sketched others, hoping that this will clarify the exposition. After introducing extreme, exposed and strongly exposed points, we end the introduction with definitions of the Hardy-spaces in which we study our problem. These are: the standard H 1 (][))) of the disc ][)) in C, H 1 (0) of a domain of finite connectivity n C C, the Bergman space A 1 (][))) of the disc and H 1 (lllln) of the unit ball in In Section 2 we will study the boundary points of the unit ball of H 1 (][))). Here good function theoretic descriptions of extreme and strongly exposed points are possible. Exposed points seem to escape such a description. The ball of H 1 (0) is the topic of Section 3. We will see that results for the disc in general do not extend to this case. In Section 4 we will turn to the Bergman space. Now all boundary points of the unit ball are exposed and many well-behaved functions, like polynomials, are strongly exposed. However, there are boundary points that are not strongly exposed. The final section is devoted to the little we know about the higher dimensional situation. The paper borrows heavily from the expository parts of the PhD-thesis of the first author, [3].

en.

1.1. Generalities. Let X be a Banach space with (only in this section) real dual space X* and let A be a bounded closed subset of X. We recall some notions that are connected with convexity properties at a boundary point of A. Research of the first author was supported by the Dutch research organization NWO.

59

©



PAUL BENEKER AND JAN WIEGERINCK

60

DEFINITION 1.1. A point x E A is called extreme if x = tp + (1- t)q for some p, q E A and 0 < t < 1 implies that p = q = x. A point x E A is called exposed if there exists L E X* such that Lx = 1 while Ly < 1 for every y E A \ { x}. The functional L is called an exposing functional for x.

For L E X* we set (1.1)

M(L,A) = sup{Lx: x E A}

and (1.2) If IlLII

(1.3)

M(A) = sup{llxll : x E A}.

= 1 and a > 0 we define a slice of A as S(L,a,A)

= {x

E A: Lx

>

M(L,A)- a}.

Let B be the closed unit ball of X. DEFINITION 1.2. A subset A of X is called dentable if for every c > 0 there is a point x E A such that x is not in the norm-closed convex hull of A\ (x + cB). Equivalently, A is dentable if for every c > 0 there is a slice S(L, a, A) of diameter less than c. DEFINITION 1.3. A point x E A C X is called a denting point if for every c > 0 there exists a slice S(L, a, A) of diameter less than c that contains x. A point x E A c X is called a strongly exposed point if there exists L E X* such that for every c > 0 there exists a > 0 such that the slice S(L, a, A) contains x and has diameter less than c. In other words, there exists LEX* such that Lx = M(L, A) and if Lxn-+ M(L, A) for {xn} C A, then Xn-+ x in the norm of X.

With these definitions at hand we make some observations. A strongly exposed point in A is exposed. Normalizing the functional L in the previous definition will yield an exposing functional for x. An exposed point of A is extreme and an extreme point of A belongs to the boundary of A. A theorem of Phelps, [38], states that if every bounded subset of X is dentable, then every bounded closed convex set is the convex hull of its strongly exposed points. It is known that separable dual spaces have the property that every bounded set is dentable, cf. [10], Chapter 6. However, neither Phelps' theorem nor its proof gives us any information about, say, which of the exposed points of a convex set C are strongly exposed points. In the rest of the paper it will be more convenient to use an equivalent, more analytic definition of strongly exposed point: An exposed point x of the set A C X with exposing functional L is strongly exposed if and only if every sequence (xn)in A with the property that Lxn -+ 1 converges strongly to x. Moreover, we will work over C, so that the definition of exposed point becomes slightly adapted. A point x E A is an exposed point of A, if there exists a (complex) functional L on X with the following property: Lx = 1 and for all y E A, y -1= x ====? Re L(y) < 1. (For A E C, ReA denotes the real part of A.) Again we say that the functional Lis an exposing functional for x, or simply that L exposes the point x.


61

THE BOUNDARY OF THE UNIT BALL IN H 1 -TYPE SPACES

1.2. Hardy spaces. Next we briefly recall the definition and some properties of Hardy spaces. Excellent introductions are [12, 16, 24], and for domains of finite connectivity [13]. We will denote the space of holomorphic functions on a domain Dc by H(D). Let ][} be the unit disc in e and 1l' its boundary.

en

DEFINITION 1.4. Let 0 < p < oo. The Hardy space HP = HP(Il}) consists of all holomorphic functions f on ][} for which

llfll~p

:= sup

O.. IFI·I-). 2n 0

1

Therefore, we can choose >.. in such a way that Re ( >.. J02 7r IFI · I quently, llf ± gll1 = 1, but g =/= 0, so f is not extreme.

g!)

= 0. ConseD

EXAMPLE 2.4. Normalized polynomials without zeros in[]) are extreme points.

2.2. Exposed points in B(H 1([]))), function theoretic methods. As far as the authors know, there is no clearcut function theoretic characterization of exposed points of B ( (H 1([]))). The necessary and sufficient condition of Helson, [21] given below in Theorem 2.8 probably comes closest. The following lemma identifies the exposing functionals. LEMMA 2.5. Iff is exposed in B(H 1([]))), then Lt:

g jg TJT2n J de f---+

is the unique exposing functional.

PROOF. Suppose

f

is an exposed point of B(H 1([]))) with exposing functional

L. By the Hahn-Banach theorem there exists a function
0. When the spaces P and Q are at positive angle, the projection N

N

-N

0

P+(L=aneiniJ) = Lanein!J, which is densely defined on L 2 (J.t), extends to a bounded operator on L 2 (J.t). Conversely, if the intersection of P and Q in L 2 (J.t) is trivial, then the definition of P+ acting on trigonometric polynomials is well-defined and extends to a bounded operator on L 2 (J.t) if and only if P and Q are at positive angle. DEFINITION 2.15. We say that a function w;:::: 0 on'][' is a Helson-Szego weight (on 1l') if there exist real valued u,v E L 00 (1l') with llvlloo < "i such that w = eu+V. (Here v is the boundary function of the harmonic conjugate of the Poisson integral of v to [)).) The following theorem of H. Helson & G. Szego elegantly describes all measures J.t for which P and Q are at positive angle. THEOREM 2.16 ([19]). The subspaces P and Q are at positive angle in L 2 (J.t) if and only if the measure J.t is of the form df..t = wd(), for some Helson-Szego weight won 1l'. COROLLARY 2.17. If the function f is strongly exposed in the unit ball of H 1 , then

lfl

is a Helson-Szego weight on']['.

PROOF. Assume f is an exposed point, such that lfl is not a Helson-Szego weight. We will show that f is not strongly exposed. Let f..t be the probability measure If I By the theorem of Helson and Szego the spaces P and Q are at zero angle (p = 1). Thus we can find sequences (Pn) and (qn) in the L 2 (J.t)-unit balls of P and Q respectively, such that

g!.

(2.4)

r" Pnqn

Jo

dj.t

---+

1,

as n ---+ oo. For n = 1, 2, ... , let fn be the H 1 -function Pnqnf· These functions are contained in the unit ball of H 1 :


THE BOUNDARY OF THE UNIT BALL IN If we set 0, the functions f and 1/ f are contained in Hl+E. The corollary together with Theorem 2.13 show that iff E B(H 1 (IDl)) satisfies the distance condition in Theorem 2.13, then f is strongly exposed if and only if 1/ f E H 1 . (It is worth noting that there exist exposed points f E H 1 such that 1/ f is in no HP, p > 0, [37, 44].) Since f and 1/ fare outer functions, the corollary is immediate from the following lemma (which is actually an exercise in [16]). LEMMA 2.19. If the function w is a Helson-Szego weight on 1!', then for all small E: > 0, we have wE Ll+E('li') and 1/w E Ll+E('li'). LEMMA 2.20. (Cf. [16], Lemma IV.3.3, p. 148.) If '1/J is a measurable real function, then the L= -distance of e-i'I/J to H 00 is less than 1 if and only if there exist E: > 0 and h E H 00 such that (2.5)

lhl2

E:

and

1'1/J+arghl:::;

2 -E: 1f

(mod 27r)

almost everywhere on 1!'.

THEOREM 2.21 ([2, 31]). Let f be an extreme point of the unit ball of H 1 . Then f is strongly exposed if and only if lfl is a Helson-Szego weight. PROOF. ([31]) By Corollary 2.17 we need only to prove the backward implication. Let us assume that the outer function f is such that lfl is a Helson-Szego weight on 1!', say, lfl = exp(u + v), where llvll= < ~· We remark that the function is exposed by Theorem 1 and Lemma 2.19. Next, again using the fact that f is outer, we notice that f(z) = eu(z)+iu(z). ev(z)-iv(z)' because the right hand side is an outer function with the appropriate absolute values on 1!'. We set
0 as lzl ---> 1 is a closed subspace of B, called the little Bloch space Bo.

Let Co denote the continuous functions on~ that are zero on 'll'. We have the following theorem of R. Coifman, R. Rochberg and G. Weiss: THEOREM 4.2 ([8]). The Bergman projection P maps L 00 (~) B. Furthermore, P maps both C(D) and Co boundedly onto B 0 •

boundedly onto

The result proved in [8] is much more general than Theorem 4.2. As we state it, the theorem may be found in [18], Theorem 1.12, with an elementary proof. There we also find the following results THEOREM 4.3. (1) The dual space of A 1 is the Bloch space B under the following pairing: g E B: f E A1 ~lim

(4.4)

{ fr(z)g(z)dA(z).

rjl }[)

(2) The dual space of the little Bloch space Bo is the Bergman space A1 under the pairing: f E A1 : g E Bo

~limrj1 }[){ f(z)gr(z)dA(z).

REMARK 4.4. When one identifies (A 1)* with the Bloch space in Theorem 1 B, the dual norm on B yields a norm that is equivalent with, but not equal to the norm 11-IIB that we have previously defined on B. Hence, there exists a norm 11-11. on the Bergman space A 1 that is equivalent to ll-ll1 and is such that the dual norm of g E B = (A 1) * equals II g II B. The strongly exposed points in the unit ball of A 1 with the norm 11-11. have been described by C. Nara ([32]), who also showed that up to isometrical isomorphisms, A1 with the 11-11• norm is the unique pre-dual of B. 4.2. Strongly exposed points of A 1 (~). The following theorem is a consequence of Theorems 5.3 and 5.2, the fact that A 1 (~) may be identified with the subspace of H 1(Jffi 2 ) consisting of functions that depend only on one variable, and the fact that these functions are exposed in B(H 1(lffi2). We set (4.5)


THE BOUNDARY OF THE UNIT BALL IN

H 1 -TYPE

SPACES

79

THEOREM 4.5. Let f E A 1 be of unit norm. Then f is strongly exposed in Ball( A 1 ) if and only if the L 00 -distance off llfl to the space (A 1 )j_ + C(D) is less than one. Henceforth we will simply write (A 1 )_i

+C

instead of (A 1 )_i

+ C(D).

The question now is: how can we estimate the distance in L 00 of r..p = f 11!1 to + C, where f is a given function in A 1 ? (Clearly the distance cannot exceed one.) Let us first look at polynomials of a particularly simple form: f (z) = c( z- o:) n, where c is normalizing. We will assume n ~ 1 because the constant functions are strongly exposed by Theorem 4.5. We distinguish three cases in order of increasing difficulty: lo:l > 1, lo:l < 1 and lo:l = 1. (A 1 )j_

The case where lo:l > 1 is very easy: f llfl is continuous on D, so f is strongly exposed. In fact, we may even take non-integer powers n and products of such functions and we always obtain strongly exposed points after normalization. When lo:l < 1, let us write r..p = f llfl = '1/Jo + 'I/J1, where 'I/J1 is compactly supported in][)) and r..p = 'lj; 1 on a neighborhood of o:, while '1/Jo is smooth on D. From (4.1) we see that P'lj; 1 is holomorphic across the unit circle because 'lj; 1 is compactly supported in ][)). Next, because '1/Jo is smooth on D, also P'lj; 0 is smooth on D. Hence Pr..p is continuous on D. Now r..p- Pr..p is bounded, so r..p = (r..p- Pr..p) + Pr..p is contained in (A 1 )_i + C. By Theorem 4.5 f is strongly exposed. Again, our reasoning readily shows that the normalized product f of functions (z- ai)ni, for all ni and all o:i tfc 'll', is strongly exposed. Now suppose lo:l = 1; we may take o: = 1. Let us write fn(z) = cn(1- z)n and 'Pn = fnllfnl· Introducing polar coordinates and applying Cauchy's theorem one finds that the exposing functional L for h is given by

L(g)

j1- z;: dA(z) = { g(z) (1- z) dA(z) = Cog(O) + C1g'(O). z J~ 1- z

= { g(z) 1 J~

But then there exists a polynomial P2 such that L(g) = I~ gp 2 dA. Therefore r..p 2 - p 2 is contained in (A 1 )j_, hence r..p 2 E (A 1 )j_ + C so h is strongly exposed. Similarly, for all even n, 'Pn is contained in (A 1 )_i +C and fn is strongly exposed in A 1 . Again, we may introduce non-integer exponents. Let ff3 = c(3(1 - z )f3, where (3 > -2 to ensure that f (3 E A 1 ; the constant Cf3 > 0 is normalizing. Set

'P(3 = ff311ff31·

PROPOSITION 4.6 ([5]). For all {3 > -1, the L 00 -distance ofr..pf3 to (A 1 )_l + C is at most Isin( {321r) I· In particular, for all (3 > -1, (3 -1- 1, 3, 5, ... , the function f f3

is strongly exposed in the unit ball of A 1 .

The proof consists of observing that if lf3- 2nl < 1, then II'Pf3 - 'P2nlloo = Isin( 7rf) I < 1. We will come back to odd exponents in Section 4.4 and end this section with and example of a boundary points of B(A 1 (][)))) that is not strongly exposed. EXAMPLE 4.7 ([5]). The normalized function f(z) strongly exposed in the unit ball of A 1 . The functions However, limf3l-2 I~ ff3Zf5 dA = 1.

2

= (1 -z) ~~:2( 1 -z)

f f3


is not tend pointwise to 0.

PAUL BENEKER AND JAN WIEGERINCK

80

4.3. The space (A 1 )j_ +C. We have observed that (A 1 )j_ + C plays the same role in Theorem 4.5 with respect to the Bergman space as (H 1 )j_ + C(1l') = H 00 +C(1l') with respect to the Hardy space H 1 ([))) (Theorem 2.13). We mentioned already in Section 2.3 that H 00 + C is closed in L 00 • From this then it followed relatively easily that H 00 +C(1l') is in fact an algebra, cf. [41], Theorem 6.5.5. How far do these results extend to the space (A 1 )j_ + C? From [5] we quote (1) A 1 ([)))j_ + C(li)) is a closed subspace of L 00 (][))). (It equals p- 1 (80 )!) (2) A 1 ([)))j_ + C(li)) is a C(li))- module. (3) A 1 ([)))j_ + C(li)) is not an algebra. (4) The space (A 1 ) j_ + C is invariant under composition with holomorphic automorphisms of[)).

EXAMPLE 4.8. [5] As for (3), let ff3 = (1- z)f3 and let 'Pf3 = ff3/lff31 for (3 E JR. Then cp 2 and cp_ 4 E (A 1 )j_, but cp_ 2 is not contained in (A 1 )j_ +C. The space (A 1 )j_ + C is not an algebra. The properties of (A 1 ) j_ + C lead in a straightforward way to the following proposition. PROPOSITION 4.9. Let f be a strongly exposed point in A 1 . Then (a) ifu is an automorphism oj][)), then the normalized function F 1 = C 1 (fou) is strongly exposed; (b) if v E A(][))) is zero-free on the circle, then the normalized function F2 = C2fv is strongly exposed. Furthermore, the functions f /IJI, Fl/IF1 and F2/IF2I have the same L 00 -distance to (A 1 )j_ +C. 1

4.4. Strong exposedness of (1 - z )f3. We saw in Section 4.2 that the functions f f3 = Cf3 ( 1 - z )f3 are strongly exposed in the unit ball of A 1 for all (3 > -1 except possibly when (3 = 1, 3, 5, .... This was deduced from rather straightforward estimates of the L 00 -distances of the functions cp = f f3 /Iff31 to the space (A 1 ) j_ + C (Proposition 4.6). In [5] a much sharper result is proved.

THEOREM 4.10. For all (3 2: 0, the Bloch distance of the function Pcpf3 to Eo 4 I sin(lh )I equals -;;: {3_;§ . SKETCH OF PROOF. It is convenient to rewrite 'Pf3 as cp{3(w) = (1- w)f31 2/(1w)f312. Using the series expansions for the Bergman kernel1/(1- zw) 2 (see (4.2)), as well as for (1- w)f31 2 , and 1/(1- w)f3! 2 , we evaluate the Bergman projection Pcpf3. One obtains Pcpf3 = L:~=O Cf3,nZn, where Cf3,n

=

-(n + 1) sin(l3f) ~ f(m + ~)r(m 21r

It is proved in [5] that for fixed (3 > 0:

fo 00

(4.6)

+ n- ~) m!(m + n + 1)! ·

L.....

r(m+~)r(m+n-~) m!(m + n + 1)!

m=O

4

= n2(3((3 + 2) (1 + o(1)),

where the o(1)-term tends to zero as n--+ oo. This implies that -2 sin( {321r)

C(3,n

= 1r((3 + 2)n (1 + o(1)),


THE BOUNDARY OF THE UNIT BALL IN

H

1

-TYPE SPACES

81

where the o(1)-term vanishes as n--+ oo. But then, , 4lsin(/37r)l lim 1(1- x 2)(Pcp) (x)l = ({3 2 ) , xil 1r +2 so the Bloch distance of Pcp 13 to 8 0 is at least 41 ;~;~~ll. large N,

On the other hand, for

~ c zn)'l < ~ nlc l·lzln-1 < 21 sin(~)l1 I( L.... j3,n - L.... j3,n 1r({3 + 2) n=N

n=N

+ o(1) 1 - lzl '

where the o(1)-term tends to zero as N increases. Using the fact that the polynomials are contained in 8 0 it follows that the Bloch distance of Pcp;3 to 8 0 is at most 4lsin(l3f)l 0 7r(/3+2) 0

CoROLLARY 4.11. Let d(cp 13 , (A 1)j_ +C) denote the L 00 -distance of cp 13 to (A 1)j_+C. Thenforallf320, (4.7)

~I sin(~)l 2

f3 + 2

< d( (A 1)_1_ C{Jj3,

+C) Y, then T is completely contractive ifTn is contractive for all n EN, where Tn is the map [xij] f---7 [T(xij)]. Similarly Tis completely isometric if II[T(xij)JII = ll[xij]ll for all n EN and [xij] E Mn(X). It is an easy exercise (using one of the common expressions for the operator norm of a matrix in Mn = Mn(C)) to prove that a linear map T : X ----> Y between 1991 MatJ;Lematics Subject Classification. Primary 46107, 46105, 47130; Secondary 46J10. This research was supported in part by a grant from the National Science Foundation.

85

©



86

DAVID P. BLECHER AND DAMON M. HAY

subspaces of C(K) spaces is completely contractive if and only if it is contractive. Consequently such a T is isometric if and only if it is completely isometric. The identification of the term 'noncommutative Banach space' with 'operator space' may be thought of as a relatively recent entry in the well known 'dictionary' translating terms between the 'commutative' and 'noncommutative' worlds. We spend a paragraph describing some other entries in this dictionary. Although these items are for the most part well known to the point of being tedious, it will be helpful to collect them here for the dual purpose of establishing notation, and for ease of reference later in the paper. The most well known item is of course the fact that the noncommutative version of a C(K) space is a unital C*-algebra B. The noncommutative version of a unimodular function in C(K) is a unitary u E B (i.e. u*u = uu* = 1). The noncommutative version of a function algebra A C C(K) 'containing constant functions' is a closed subalgebra A of a C*-algebra B, with 18 E A. We call such A a unital operator algebra. For a unital subset Sofa C*algebra B, we will take as a simple noncommutative version of the assertion 'S C C(K) separates points of K', the assertion 'the C*-subalgebra of B generated by S (namely, the smallest C*-subalgebra of B containing S) equals B'. The analogue of a closed subset E of a compact set K is a quotient B /I, where I is a closed twosided ideal in a unital C*-algebra B. More generally, unital *-homomorphisms 1r between unital C* -algebras are the noncommutative version of continuous functions T between compact spaces. Indeed clearly any such T : K 1 ---+ K 2 gives rise to the unital *-homomorphism C(K2 ) ---+ C(KI) of 'composition with r', and conversely it is not much harder to see that any unital *-homomorphism C(K2 ) ---+ C(K1) comes from a continuous Tin this way. Moreover such 1r is 1-1 (resp. onto) if and only if the corresponding T is onto (resp. 1-1). Thus the noncommutative version of a homeomorphism between compact spaces is a (surjective 1-1) *-isomorphism between unital C*-algebras. Coming back to 'noncommutative functional analysis', it is convenient for some purposes (but admittedly not for others) to view 'complete isometries' as the noncommutative version of isometries. It is very important in what follows that a 1-1 *-homomorphism 1r: A---+ B between C*-algebras, is by a simple and well known spectral theory argument, automatically an isometry, and consequently (by the same principle applied to 7rn), a complete isometry. Similarly, a *-homomorphism 1r : A ---+ B (which is not a priori assumed continuous) is automatically completely contractive, and has a closed range which is a C* -algebra *-isomorphic to the C* -algebra quotient of A by the obvious two-sided ideal, namely the kernel of the *-homomorphism. The entries we have just described in this 'dictionary' are all easily justified by well known theorems (for example Gelfand's characterization of commutative C*algebras). That is, if one applies the noncommutative definition in the commutative world, one recovers exactly the classical object. Similarly one sometimes finds oneself in the very nice 'ideal situation' where one can prove a theorem or establish a theory in the noncommutative world (i.e. about operator spaces or operator algebras), which when one applies the theorem/theory to objects which are Banach spaces or function algebras, one recovers exactly the classical theorem/theory. An illustration of this point is the Banach-Stone theorem. The following is a much simpler form of Kadison's characterization of isometries between C* -algebras [17]:


COMPLETE ISOMETRIES- AN ILLUSTRATION

87

THEOREM 1.1. (Folklore) A surjective linear map T : A --+ B between unital C* -algebras is a complete isometry if and only if T = u1r( ·), for a unitary u E B and a *-isomorphism 1r: A--+ B. PROOF. (Sketch.) The easy direction is essentially just the fact mentioned earlier that 1-1 *-homomorphisms are completely isometric. The other direction can be proved by first showing (as with Kadison's theorem) that T(1) is unitary, so that without loss of generality T(1) = 1. The well known Stinespring theorem has as a simple consequence the Kadison-Schwarz inequality T(a)*T(a):::; T(a*a). Applying this to r- 1 too yields T(a)*T(a) = T(a*a), and now the result follows immediately from the 'polarization identity' a*b = E~= 0 (a + ikb)*(a + ikb). D

t

Note that if one takes A = C(K1 ) and B = C(K2 ) in Theorem 1.1, and consults the 'dictionary' above, then one recovers exactly the classical Banach-Stone theorem. Indeed as we remarked earlier, in this case complete isometries are the same thing as isometries, unitaries are unimodular functions, and a *-isomorphism is induced by a homeomorphism between the underlying compact spaces. Indeed consider the following generalization of the Banach-Stone theorem: THEOREM 1.2. [15, 22, 1, 20] Let n be compact and Hausdorff, and A a unital function algebra. A linear contraction T : A --+ C(O) is an isometry if and only if there exists a closed subset E of n, and two continuous functions 'Y : E --+ '][' and tp : E --+ aA, with tp surjective, such that for all y E E

T(f)(y) = 'Y(Y)f(tp(y)). Here aA is the Shilov boundary of A (see Section 2). We have supposed that T maps into a 'selfadjoint function algebra' C(O); however since any function algebra is a unital subalgebra of a 'selfadjoint' one, the theorem also applies to isometries between unital function algebras. If A is a C(K) space too, then aA = K and then the theorem above is called Holsztynski's theorem. We refer the reader to [16] for a survey of such variants on the classical Banach-Stone theorem. Often the transition from the 'classical' to the 'noncommutative' involves the introduction of much more algebra. Next we appeal to our dictionary above to give an equivalent restatement of Theorem 1.2 in more algebraic language. THEOREM 1.3. (Restatement of Theorem 1.2) Let A, B be unital function algebras, with B selfadjoint. A linear contraction T : A --+ B is an isometry if and only if

(A) there exists a closed ideal I of B, a unitary u in the quotient C* -algebra B I I, and a unitall-1 *-homomorphism 1r : A --+ B I I, such that q1 (T( a)) = u1r(a) for all a EA. Here qi is the canonical quotient *-homomorphism B--+ B I I. In light of Theorems 1.1 and 1.3 one would imagine that for any complete isometry T : A --+ B between unital operator algebras, the condition (A) above should hold verbatim. This would give a pretty noncommutative generalization of Theorem 1.3. Indeed if Ran T is also a unital operator algebra, then this is true (see eg. B.1 in [3]). However, it is quite easily seen that such a result cannot hold generally. For example, let Mn = Mn(C); for any x E Mn of norm 1, the map .>. f---> >.x is a complete isometry from C into Mn. Now Mn is simple (i.e. has no


88


nontrivial two-sided ideals), and so if the result above was valid then it follows immediately that x = u. This is obviously not satisfactory. To resolve the dilemma presented in the last paragraph, we have offered in [5] several alternatives. For example, one may replace the quotient B I I by a quotient of a certain *-subalgebra of B. The desired relation q1 (T(a)) = u1r(a) then requires u to be a unitary in a certain C* -triple system (by which we mean a subspace X of a C*-algebra A with XX* XC X). Or, one may replace the quotient BII by a quotient B I (J + J*), where J is a one-sided ideal of B. Such a quotient is not an algebra, but is an 'operator system' (such spaces have been important in the deep work of Kirchberg (see [18, 19] and references therein). Alternatively, one may replace such quotients altogether, with certain subspaces of the second dual B** defined in terms of certain orthogonal projections of 'topological significance' (i.e. correspond to characteristic functions of closed sets in K if B = C(K)) in the second dual B** (which is a von Neumann algebra [25]). The key point of all these arguments, and indeed a key approach to Banach-Stone theorems for linear maps between function algebras, C* -algebras or operator algebras, is the basic theory of C* -triple systems and triple morphisms, and the basic properties of the noncommutative Shilov boundary or triple envelope of an operator space. These important and beautiful ideas originate in the work of Arveson, Choi and Effros, Hamana, Harris, Kadison, Kirchberg, Paulsen, Ruan, and others. Indeed our talk at the conference spelled out these ideas and their connection with the Banach-Stone theorem; and the background ideas are developed at length in a book the first author is currently writing with Christian Le Merdy [7] (although we do not characterize non-surjective complete isometries there). Moreover, a description of our work from this perspective, together with many related results, may be found in [12]. Thus we will content ourselves here with a survey of some related and interesting topics, and with a new and self-contained proof of some characterizations of complete isometries between unital operator algebras which do not appear elsewhere. This proof has several advantages, for example the projections arising naturally with this approach seem to be more useful for some purposes. Also it will allow us to avoid any explicit mention of the theory of triple systems (although this is playing a silent role nonetheless). We also show how such noncommutative results are generalizations of the older characterizations of into isometries between function algebras or C(K) spaces. We thank A. Matheson for telling us about these results. In the final section we present some evidence towards the claim that (general) isometries between operator algebras are not the correct noncommutative generalization of isometries between function algebras. For the reader who wants to learn more operator space theory we have listed some general texts in our bibliography. 2. The noncommutative Shilov boundary At the present time the appropriate 'extreme point' theory is not sufficiently developed to be extensively used in noncommutative functional analysis. Although several major and beautiful pieces are now in place, this is perhaps one of the most urgent needs in the subject. However there are good substitutes for 'extreme point' arguments. One such is the noncommutative Shilov boundary of an operator space. Recall that if X is a closed subspace of C(K) containing the identity function lx on


COMPLETE ISOMETRIES- AN ILLUSTRATION

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K and separating points of K, then the classical Shilov boundary may be defined to be the smallest closed subset E of K such that all functions f E X attain their norm, or equivalently such that the restriction map f ~---+ f1E on X is an isometry. This boundary is often defined independently of K, for example if A is a unital function algebra then we may define the Shilov boundary as we just did, but with K replaced by the maximal ideal space of A. In fact we prefer to think of the classical Shilov boundary of X as a pair (aX, i) consisting of an abstract compact Hausdorff space aX, together with an isometry j: X---+ C(aX) such that j(lK) =lax and such that j(X) separates points of ax, with the following universal property: For any other pair (0, i) consisting of a compact Hausdorff space nand a complete isometry i : X---+ C(n) which is unital (i.e. i(lK) = lA), and such that i(X) separates points off!, there exists a (necessarily unique) continUOUS injection T : aX ---+ f2 SUCh that i(x)(r(w)) = j(x)(w) for all X E X,w E ax. Such a pair (aX,i) is easily seen to be unique up to an appropriate homeomorphism. The fact that such ax exists is the difficult part, and proofs may be found in books on function algebras (using extreme point arguments). Consulting our 'noncommutative dictionary' in Section 1, and thinking a little about the various correspondences there, it will be seen that the noncommutative version of this universal property above should read as follows. Or at any rate, the following noncommutative statements, when applied to a unital subspace X C C(K), will imply the universal property of the classical Shilov boundary discussed above. Firstly, a unital operator space is a pair (X, e) consisting of an operator space X with fixed element e E X, such that there exists a linear complete isometry "from X into a unital c* -algebra C with "-(e) = lc. A 'noncommutative Shilov boundary' would correspond to a pair (B,j) consisting of a unital c*-algebra B and a complete isometry j : X ---+ B with j(e) = lB, and whose range generates B as a c* -algebra, with the following universal property: For any other pair (A, i) consisting of a unital c* -algebra and a complete isometry i : X ---+ A which is unital (i.e. i( e) = lA), and whose range generates A as a c* -algebra, there exists a (necessarily unique, unital, and surjective) *-homomorphism 1r : A ---+ B such that 1r o i = j. Happily, this turns out to be true. The existence for any unital operator space (X, e) of a pair (B, j) with the universal property above is of course a theorem, which we call the Arveson-Hamana theorem [2, 13] (see [3] for complete details). As is customary we write c;(X) forB or (B,j), this is the 'C*-envelope of X'. It is essentially unique, by the universal property. If X = A is a unital operator algebra (see Section 1 for the definition of this), then j above is forced to be a homomorphism (to see this, choose an i which is a homomorphism, and use the universal property). Thus A may be considered a unital subalgebra of c;(A). If A is already a unital c* -algebra, then of course we can take c;(A) =A. To help the reader get a little more comfortable with these concepts, we compute the 'noncommutative Shilov boundary' in a few simple examples.

Example 1. Let Tn be the upper triangular n x n matrices. This is a unital subspace of Mn, and no proper *-subalgebra of Mn contains Tn. Let (B,j) be the C* -envelope of Tn. By the universal property of the C* -envelope, there is a surjective *-homomorphism 1r : Mn ---+ B such that n(a) = j(a) for a E Tn. The kernel of 1r is a two-sided ideal of Mn. However Mn has no nontrivial two-sided ideals. Hence 1r is 1-1, and is consequently a *-isomorphism, and we can thus identify Mn with B. Thus Mn is a C*-envelope of Tn.


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Example 2. Consider the linear subspace X of M 3 with zeroes in the 1-3, 2-3, 2-1, 3-1 and 3-2 entries, and with arbitrary entries elsewhere except for the 3-3 entry, which is the average of the 1-1 and 2-2 entries. It is easy to see that the C* -algebra generated by X inside M 3 is M 2 EB C. However this is not the C* -envelope. Indeed the 3-3 entry here is redundant, since the norm of x E X is the norm of the upper left 2 x 2 block of x. The canonical *-homomorphism from M 2 EB C onto M 2 when restricted to X is a unital complete isometry from X onto T2 (see Example 1). Thus if one takes the quotient of M 2 EB C by the kernel of this homomorphism, namely the ideal 02 EB C, then one obtains M 2 , which by Example 1 is the C* -envelope. Indeed this is typical when calculating the C* -envelope of a unital subspace X of Mn. The C* -algebra generated by X is a finite dimensional unital C* -algebra. However such a C* -algebras is *-isomorphic to a finite direct sum B of full 'matrix blocks' Mnk. Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then x f--t x(1 8 p) is completely isometric. If one eliminates such blocks then the remaining direct sum of blocks is the C* -envelope. Example 3. Let B be a unital C*-algebra. Consider the unital subspace S(B) of the C* -algebra M 2 (B) consisting of matrices

[~; :1 ]

for all x, y E B and >.., J.L complex scalars. We claim that M 2 (B) is the C* -envelope C of S(B), and we will prove this using a similar idea to Example 1 above. Namely, first note that M 2 (B) has no proper C*-subalgebra containing S(B), Thus by the Arveson-Hamana theorem there exists a *-homomorphism 1r : M 2 (B) ----> C which possesses a property which we will not repeat, except to say that it certainly ensures that 1r applied to a matrix with zero entries except for a nonzero entry in the 1-2 position, is nonzero. It suffices as in Example 1 to show that Ker 1r = {0}. Suppose that 1r(x) = 0 for a 2 x 2 matrix x E M 2 (B). Let Eij be the four canonical basis matrices for M 2 , thought of as inside M 2 (B). Then 1r(E1ixEj 2 ) = 7r(Eli)7r(X)7r(Ej 2 ) = 0 for i, j = 1, 2. Thus by the fact mentioned above about the 1-2 position, we must have E 1 ;xEj 2 = 0. Thus x = 0. In fact a variant of the C* -envelope or 'noncommutative Shilov boundary' can be defined for any operator space X. This is the triple envelope of Hamana (see [14]). This is explained in much greater detail in [3], together with many applications. For example it is intimately connected to the 'noncommutative M-ideals' recently introduced in [4]. This 'noncommutative Shilov boundary' is, as we mentioned in Section 1, a key tool for proving various Banach-Stone type theorems. However in the present article we shall only need the variant described earlier in this section. 3. Complete isometries between operator algebras

We begin this section with a collection of very well known and simple facts about closed two-sided ideals I in a C* -algebra A, and about the quotient C* -algebra A/ I. We have that Ij_j_ is a weak* closed two-sided ideal in the von Neumann algebra A**, and there exists a unique orthogonal projection e in the center of A** with Ij_j_ = A**(1- e). The projection 1- e is called the support projection for I, and


COMPLETE ISOMETRIES - AN ILLUSTRATION

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1- e may be taken to be the weak* limit in A** of any contractive approximate identity for I. Thus it follows that A**/ I 1_1_ ~ A** e as C* -algebras. Therefore also A/I

c

(A/I)**~

A** /Il_l_ ~ A**e

as C*-algebras. Explicitly, the composition of all these identifications is a 1-1 *homomorphism taking an a+ I in A/ I' to ae = eae in A**. Here is the canonical embedding A -+ A** (which we will sometimes suppress mention of). Thus A/ I may be regarded as a C* -subalgebra of A**, or of the C* -algebra eA **e. We next illustrate the main idea of our theorem with a simple special case. (The following appeared as part of Corollary 3.2 in the original version of [5], with the proof left as an exercise). Suppose that T : A -+ B is a complete isometry between unital C*algebras, and suppose that Tis unital too, that is T(1) = 1. Let C be the C*-subalgebra of B generated by T(A). Applying the Arveson-Hamana theorem 1 we obtain a surjective *-homomorphism(): C-+ A such that ()(T(a)) =a for all a E A. If I is the kernel of the mapping (), then C /I is a unital C* -algebra *-isomorphic to A. Indeed there is the canonical *-isomorphism 'Y : A -+ C /I induced by (), taking a to T(a) +I. The next point is that C /I may be viewed as we mentioned a few paragraphs back, as a C* -subalgebra of C**, and therefore also of B**. Indeed if e is the central projection in C** mentioned there, then C /I may be viewed as a C* -subalgebra of eC** e C eB** e C B**. In view of the last fact, the map"( induces an 1-1 *-homomorphism 7f: A-+ B** taking an element a E A to the element of B** which equals A

-

(1)

eT(a)e

-

(these are equal because e is central inC**). Conversely, if T: A-+ B is a complete contraction for which there exists a projection e E B** such that eT(a)e is a 1-1 *-homomorphism 1r, then for all a E A,

IIT(a)ll ~ lleffaJell = 117r(a)ll = llall using the fact mentioned earlier that 1-1 *-homomorphisms are necessarily isometric. Thus T is an isometry, and a similar argument shows that it is a complete isometry. Thus we have characterized unital complete isometries T: A-+ B. If His a Hilbert space on which we have represented the von Neumann algebra B** as a weak* closed unital *-subalgebra, then B may be viewed also as a unital C*-subalgebra of B(H), whose weak* closure in B(H) is (the copy of) B**. In this case we shall say that B is represented on H universally. (The explanation for this term is that the well-known 'universal representation' 1fu of a C*-algebra is 'universal' in our sense, and conversely if 7f is a representation which is 'universal' in our sense then 1r(B)" is isomorphic to 7ru(B)" ~ B**. See [27] Section 1.) If, further, e E B** is a projection for which (1) holds, then with respect to the splitting H = eH EB (1- e)H we may write T(a)

=

[ 7r(.) 0

0 ] S(-) '

for all a E A. We will see that this is essentially true even if T(1A) of- 18

:

1We remark in passing that one does not need the full strength of the Arveson-Hamana theorem here, one may use the much simpler [8] Theorem 4.1.


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THEOREM 3.1. Let T : A ---+ B be a completely contractive linear map from a unital operator algebra into a unital c* -algebra. Then the following are equivalent:

(i) T is a complete isometry, (ii) There is a partial isometry u E B** with initial projection e E B**, and a (completely isometric) 1-1 *-homomorphism 1r : c;(A)

1r(l) = e, such that for all a E A T(a)e

---+

eB**e with

= u1r(a) and 1r(a) = u*T(a).

Moreover e may be taken to be a 'closed projection' (see [25] 3.11, and the discussion towards the end of our proof). (iii) If H is a Hilbert space on which B is represented universally, then there exist two closed subspaces E, F of the Hilbert space H, a 1-1 *-homomorphism 1r: c;(A) ---+ B(E) with 1r(l) =IE, and a unitary u: E---+ F, such that T(a)IE

= u1r(a),

and T(a)IE.1 C Fj_, for all a EA. Here Ej_ for example is the orthocomplement of E in H. (iv) If H is as in (iii), then there exists two closed subspaces E, F of H, a unital 1-1 *-homomorphism 1r : c;(A) ---+ B(E), a complete contraction S : c; (A) ---+ B(Ej_, Fj_), and unitary operators U : E \fJ Fj_ ---+ H and V : H ---+ E \fJ Ej_, such that 0 S(a)

Jv

for all a EA.

(v) There is a left ideal J of B, a 1-1 *-homomorphism 1r from C;(A) into

a unital subspace of B I (J + J*) which is a C* -algebra, and a 'partial isometry' u in B I J such that qJ(T(a))

= u1r(a)

&

1r(a) = u*qJ(T(a))

for all a E A, where QJ is the canonical quotient map B

---+

B I J.

Before we prove the theorem, we make several remarks. First, we have taken B to be a C* -algebra; however since any unital operator algebra is a unital subalgebra of a unital C* -algebra this is not a severe restriction. We also remark that there are several other items that one might add to such a list of equivalent conditions. See [5, 6]. Items (ii)-(iv), and the proof given below of their equivalence with (i), are new. We acknowledge that we have benefitted from a suggestion that we use the Paulsen system to prove the result. This approach is an obvious one to those working in this area (Ruan and Hamana used a variant of it in their work in the '80's on complete isometries and triple morphisms [28, 14]). However we had not pushed through this approach in the original version of [5] because this method does not give several of the results there as immediately. Statement (v) above has been simply copied from [5, 6] without proof or explanation. We have listed it here simply because Theorem 1.3 may be particularly easily derived from it as the special case when A and Bare commutative (see comments below). Note that (iii) above resembles Theorem 1.2 superficially.



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PROOF. The fact that the other conditions all imply (i) is easy, following the idea in the paragraph above the theorem, namely by using the fact that a 1-1 *-homomorphism is completely isometric. In the remainder of the proof we suppose that T is a complete isometry. We view A as a unital subalgebra of c;(A) as outlined in Section 3. We define a subset S(B) of M 2(B) as in Example 3 in Section 2. Similarly define a subset S(T(A)) of S(B) using a similar formula (note that S(T(A)) has 1-2 entries taken from T(A) and 2-1 entries taken from T(A)*). Similarly we define the subset S(A) of the C*-algebra M 2(C;(A)) (i.e. S(A) has scalar diagonal entries and off diagonal entries from A and A*). We write 1 EB 0 for the matrix in S(A) with 1 as the 1-2 entry and zeroes elsewhere. Similarly for 0 EB 1. We also use these expressions for the analogous matrices in S(B). The map : S(A)---+ S(T(A)) C M 2(B) taking [ Al y*

x ]

pJ

[

).1 T(y)*

f---7

T(x) ] ttl

is well known to be a unital complete isometry (this is the well known Paulsen lemma, see the proof of 7.1 in [23]). Let C be the C* -subalgebra of M 2(B) generated by S(T(A)). The C*-envelope of S(A) is well known to be M 2(C;(A)) (see Example 3 in Section 2 where we proved this in the case that A is already a C* -algebra, or for example [3] Proposition 4.3 or [30]). Thus by the Arveson-Hamana theorem we obtain a surjective *-homomorphism C ---+ M 2 ( (A)) such that o is simply the canonical embedding of S(A) into M 2(C;(A)). As in the special case considered above the theorem, we let Io be the kernel of the mapping e, then c I Io is a unital C*-algebra *-isomorphic to M 2 (C;(A)). Indeed there is the canonical *-isomorphism'"'!: M 2(C;(A))---+ Clio induced bye, taking

e:

[~; :1 ]

f---7

[

T~:)*

c;

T~~)

]

+

e

Io.

As in the simple case above the theorem, C I ! 0 may be viewed as a C* -subalgebra of p0 C**p0 , for a central projection p0 E C** (namely, the complementary projection to the support projection of ! 0 ). Now p0 C**p 0 C C** c M 2(B)**, and it is well known that M 2(B)** ~ M 2(B**) as C*-algebras. Thus we may think of C** as a C*-subalgebra of M 2(B**). Also, C** contains C as a C*-subalgebra, and the projections 1 EB 0 and 0 EB 1 in C correspond to the matching diagonal projections 1 EB 0 and 0 EB 1 in M 2(B**). These last projections therefore commute with p0 , since p0 is central in C**, which immediately implies that p0 is a diagonal sum fEB e of two orthogonal projections e, f E B**. Thus we may write the C* -algebra poM2(B**)po as the C*-subalgebra [ f B** f eB** f

f B**e ] eB**e

of M 2(B**). We said above that Clio may be regarded as a C*-subalgebra of the subalgebrap0 M 2 (B**)p 0 of A12(B**). Thus the map'"'( induces a 1-1 *-homomorphism \fl : M 2(C;(A)) ---+ M 2(B**). It is easy to check that \fl(l EB 0) = fEB 0 and \fi(O EB 1) = 0 EB e. Since \fl is a *-homomorphism it follows that \fl maps each of the four corners of M 2(c; (A)) to the corresponding corner of p0 M 2( B** )Po C M 2( B**). We let R : c;(A) ---+ f B**e be the restriction of \fl to the '1-2-corner'. Since \fl is 1-1, it follows that R is 1-1. If 1r is the restriction of \fl to the '2-2-corner', then 1r is a *-homomorphism C;(A)---+ eB**e taking lA to e. Applying the *-homomorphism


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\II to the identity

[~ ~][~

~]=[~

~]

we obtain that u = R(1) is a partial isometry, with u*u = 7r(1) = e. Similarly uu* =f. A similar argument shows that R(a) = R(1)7r(a) for all a E c;(A). Thus u* R(a) = u*u1r(a) = 1r(a) for all a E C;(A). Next, we observe that \II takes the matrix z which is zero except for an a from A in the 1-2-corner, to the matrix w = p0 (z)p0 . Since (z) E C** and p0 is in

--------(z)p

----p ----(z). Also

the center of that algebra, we also have w = w viewed as 0 = 0 a matrix in M 2 (B**) has zero entries except in the 1-2-corner, which (by the last sentence) equals

-----

JT(a)e = T(a)e = JT(a).

Also using these facts and a fact from the end of the last paragraph we have u*T(·)

---* ------* = R(1)*T(·) = (JT(1)e)*T(·) = eT(1) JT(·) = eT(1) T(-)e = u* R(·) = 1r.

Thus T(·)e

= JT(·) = uu*T(·) = u1r(·).

We have now also established most of (ii). One may deduce (iii) from (ii) by viewing B c B** C B(H), and setting E = eH, and F = (uu*)H. We also need to use facts from the proof above such as u*u = e. Clearly (iv) follows from (iii). As we said above, we will not prove (v) here. Claim: if e is the projection in (ii) above, then 1 - e is the support projection for a closed ideal I of a unital *-subalgebra D of B. Equivalently (as stated at the start of this section), there is a (positive increasing) contractive approximate identity (bt) for I, with bt ---+ 1 - e in the weak* topology. This claim shows that 1 - e is an 'open projection' in B**, so that e is a closed projection, as will be obvious to operator algebraists from [25] section 3.11 say. For our other readers we note that for what comes later in our paper, one can replace the assertion about closed projections in the statement of Theorem 3.1 (ii) with the statement in the Claim above. To prove the Claim, recall from our proof that Po = f EB e = 1c - p 1 , where Pl is the support projection for a closed ideal I 0 of C. Thus p 1 = (1- f) EB (1- e). As stated at the start of Section 3, p 1 is the weak* limit in C**, and hence also in M 2 (B**), of a contractive approximate identity (et) of I 0 . By the separate weak* continuity of the product in a von Neumann algebra, it follows that the net bt = (0 EB 1)et(O EB 1) has weak* limit (0 EB 1)p1 (OEB 1) = 0 EB (1- e). Viewing these as expressions in B, the above says that bt ---+ 1- e weak* in B**. View (OEB 1)C(OEB 1) as a *-subalgebra D of B, and view (0 EB 1)I0 (0 EB 1) as a two sided ideal I in D. It is easy to see that (bt) is a contractive approximate identity of I. Thus it follows that 1 - e is the support projection of the ideal I. D Some applications of results such as Theorem 3.1 may be found in [6]. Next we discuss briefly the relation between our noncommutative characterization of complete isometries (for example Theorem 3.1 above), and Theorem 1.3. Our point is not to provide another proof for Theorem 1.3 - the best existing proof is certainly short and elegant. Rather we simply wish to show that the noncommutative result contains 1.3. Indeed Theorem 1.3 quite easily follows from Theorem



95

3.1 (v). Since however we did not prove Theorem 3.1 (v), we give an alternate proof. COROLLARY 3.2. Let A, B be a unital function algebras, with B selfadjoint. Then condition (ii) in Theorem 3.1 implies condition (A) in Theorem 1.3.

By hypothesis, T(·)e = U?r(·), and u*u = e = 1r(1) so that u = u1r(1) = T(1)e. Thus eT(1)*T(·)e = u*u1r(·) = 7r(1)7r(·) = 1r, so that Ran 1r C eBe =Be (note B** is commutative in this case). From [25] 3.11.10 for example, the 'closed projection' e in B** corresponds to a closed ideal J in B whose support projection is 1 - e. Alternatively, to avoid quoting facts from [25], we will also deduce this from the 'Claim' towards the end of the proof of Theorem 3.1. If I is the ideal in that Claim, let J be the closed ideal in B generated by I. Since J = BI, the contractive approximate identity of I is a right contractive approximate identity of J. Thus J ha8 support projection 1 - e too, by the first paragraph of Section 3 above. By facts in the just quoted paragraph, we have a canonical unital 1-1 map 7] : B / J ---+ B** taking the equivalence class b + J of b E B to ebe. Indeed in this commutative case we see by inspection that 77 is a *-homomorphism from the C*algebra B/J onto the C*-subalgebra M = eBe of B**. Define B(a) = 77- 1 (1r(a)), this is a 1-1 *-homomorphism A ---+ B/ J. Since 1r(1) = e, (} is a unital map too. Since uu* = u*u = e, u is unitary in M, and so r = 77- 1 (u) is unitary in B/ J. Note also that T(a)e = 77(T(a) + J). Applying 77- 1 to the equation T(·)e = u1r(·), we obtain qJ(T(a)) = r B(a), that is, condition (A) in Theorem 1.3. D PROOF.

If one attempts to use the ideas above to find a characterization analogous to condition (A) from Theorem 1.3 but in the noncommutative case, it seems to us that one is inevitably led to a condition such as (v) in Theorem 3.1. We address a paragraph to experts, on generalizations of the proof of Theorem 3.1. Consider a complete isometry between possibly non-unital C* -algebras. Or much more generally, suppose that T is a complete isometry from an operator space X into a C* -triple system W. One may form the so called 'linking C* -algebra' of W, with the identities of the 'left and right algebras of W' adjoined. Call this £'(W). As in the proof of Theorem 3.1 we think of S(W) C £'(W). Similarly, if Z is the 'triple envelope' of X (or if X= Z is already a C*-algebra or C*-triple system), then we may consider S(X) C S(Z) C £'(Z). As in the proof of Theorem 3.1 we obtain firstly a unital complete isometry : S(X)---+ S(T(X)) C £'(Z), and then a unital1-1 *-homomorphism 1r: L.'(Z) ---+ £'(W)**. By looking at the 'corners' of 1r we obtain projections e, f in certain second dual von Neumann algebras, so that fT(·)e is (the restriction to X of a completely isometric) a 1-1 triple morphism into W**. In fact we have precisely such a result in [5] (see Section 2 there), but the key point is that the new proof gives different projections e, f, which are more useful for some purposes.

4. Complete isometries versus isometries Finally, as promised we discuss why we believe that in this setting of nonsurjective maps between c* -algebras say, general isometries are not the 'noncommutative analogue' of isometries between function algebras. The point is simply this. In the function algebra case we can say thanks to Holsztynski's theorem that the isometries are essentially the maps composed of two disjoint pieces R and S, where R is


96


isometric and 'nice', and S is contractive and irrelevant. However at the present time it looks to us unlikely that there ever will be such a result valid for general nonsurjective isometries between general c* -algebras. The chief evidence we present for this assertion is the very nice complementary work of Chu and Wong [9] on isometries (as opposed to complete isometries) T : A ----+ B between C* -algebras. They show that for such T there is a largest projection p E B** such that T( · )p is some kind of Jordan triple morphism. This appears to be the correct 'structure theorem', or version of Kadison's theorem [17], for nonsurjective isometries. However as they show, the 'nice piece' R = T(·)p is very often trivial (i.e. zero), and is thus certainly not isometric. Thus this approach is unlikely to ever yield a characterization of isometries. A good example is A = M 2 , the smallest noncommutative C* -algebra. Simply because A is a Banach space there exists, as in the discussion in the first paragraph of our paper, a linear isometry of A into a C(K) space. However it is easy to see that there is no nontrivial *-homomorphism or Jordan homomorphism from A into a commutative C* -algebra. Such an isometry is uninteresting, and this is perhaps because the interesting 'nice part' is zero. Thus we imagine that the 'good noncommutative notions of isometry' are either complete isometries or the closely related class of maps for which the piece T(-)p from [9] is an isometry. This leads to three questions. Firstly, can one independently characterize the last mentioned class? Secondly, if Tis a complete isometry, then is the projection p in the last paragraph equal (or closely related) to our projection e above? Finally, H. Pfitzner has remarked to us, there is already a gap between the isometry and the 2-isometry cases (not only isometries and complete isometries). It would be interesting if there were a characterization of 2-isometries.

References [1] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413-428. [2] W. B. Arveson, Subalgebras ofC*-algebras, Acta Math. 123 (1969), 141-224; II, 128 (1972), 271-308. [3] D. P. Blecher, The Shilov boundary of an operator space, and the characterization theorems, J. Funct. An. 182 (2001), 280-343. [4] D. P. Blecher, E. G. Effros and V. Zarikian, One-sided M -Ideals and multipliers in operator spaces. I. To appear Pacific J. Math. [5] D. P. Blecher and D. Hay, Complete isometries into C* -algebras, http:/ /front.math.ucdavis.edu/math.OA/0203182, Preprint (March '02). [6] D. P. Blecher and L. E. Labuschagne, Logmodularity and isometries of operator algebras, To appear, Trans. Amer. Math. Soc .. [7] D. P. Blecher and C. Le Merdy, Operator algebras and their modules - an operator space approach, To appear, Oxford Univ. Press. [8] M. D. Choi and E.G. Effros, The completely positive lifting problem for C* -algebras, Ann. Math. 104 (1976), 585-609. [9] C-H. Chu and N-C. Wong, Isometries between C* -algebras, Preprint, to appear Revista Matematica Iberoamericana. [10] J. B. Conway, A Course in Operator Theory, AMS, Providence, 2000. [11] E. G. Effros and Z. J. Ruan, Operator Spaces, Oxford University Press, Oxford (2000). [12] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Book to appear, CRC press. [13] M. Hamana, Injective envelopes of operator systems, Pub!. R.I.M.S. Kyoto Univ. 15 (1979), 773-785.



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[14] M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77-93. [15] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 24 (1966), 133-136. [16] K. Jarosz and V. Pathak, Isometries and small bound peturbations of function spaces, In "Function Spaces", Lecture Notes in Pure and Applied Math. Vol. 136, Marcel Dekker (1992). [17] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. [18] E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras inC* -algebras, J. Funct. An. 129 (1995), 1-34. [19] E. Kirchberg and S. Wassermann, C*- algebras generated by operator systems, J. Funct. Analysis 155 (1998), 324-351. [20] A. Matheson, Isometries into function algebras, To appear. [21] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodia Math. Sem. Rep. 11 (1959), 182-188. [22] W. P. Novinger, Linear isometries of subspaces of continuous functions. Studia Math. 53 (1975), 273-276. [23] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, London, 1986. [24] V. I. Paulsen, Completely bounded maps and operator algebras, To appear Cambridge University Press. [25] G. Pedersen, C*-algebras and their automorphism groups, Academic Press (1979). [26] G. Pisier, Introduction to operator space theory, To appear Camb. Univ. Press. [27] M. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. [28] Z. J. Ruan, Subspaces of C* -algebras, Ph. D. thesis, U.C.L.A., 1987. [29] E. L. Stout, The theory of uniform algebras, Bogden and Quigley (1971). [30] C. Zhang, Representations of operator spaces, J. Oper. Th. 33 (1995), 327-351. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON,

E-mail address, David P. Blecher: dblecher@math. uh. edu E-mail address, Damon Hay: dhay@math. uh. edu


TX 77204-3008




Some Recent Trends and Advances in Certain Lattice Ordered Algebras Karim Boulabiar, Gerard Buskes, and Abdelmajid Triki ABSTRACT. In this paper we give a survey, intended as both a supplement as well as an update to a survey by Huijsmans [57], with results that have been obtained in the last ten years on Archimedean lattice ordered algebras. Special attention is paid to /-algebras, almost /-algebras and d-algebras and problems that were posed in the survey by Huijsmans about these special classes of lattice ordered algebras.

CONTENTS

1. Introduction 2. Definitions and elementary properties 3. £-algebra multiplications in C (X) 4. Multiplication by an element as an operator 5. Uniform completion and Dedekind completion 6. Powers in £-algebras 7. Functional Calculus on !-algebras 8. Relationships between £-algebra multiplications 9. Connection between algebra and Riesz homomorphisms 10. Positive derivations 11. Cauchy-Schwarz inequalities 12. Order biduals 13. Ideal theory 14. Representation of !-algebras 15. Linear biseparating maps on !-algebras References 1991 Mathematics Subject Classification. 06F25, 13J25, 16W80, 46A40, 46B40, 46B42, 46E25, 47L07,47B47,47B65. Key words and phmses. almost /-algebra, algebra homomorphism, /-algebra, d-algebra, lattice ordered algebra, order ideal, orthomorphism, representation theory, Riesz homomorphism, ring ideal, space of continuous functions, uniformly complete Riesz space. The second named author gratefully acknowledges support from an Office of Naval Research Grant with number N00014-01-1-0322. Part of this survey was written while the first named author was visiting the University of Mississippi in the Spring of 2002.

99

©



100

KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

1. Introduction

The history of lattice ordered vector spaces (so called Riesz spaces or vector lattices) goes back to Riesz and the International Congress of Mathematicians in Bologna in 1928. A study of the most important class of lattice ordered algebras (but not their name), !-algebras, was initiated by Nakano in [75] for a--Dedekind complete ordered vector space in 1951, subsequently in 1953 by Amemiya in [3], and finally with its present definition and name in 1956 by Birkhoff and Pierce in [22]. A precise date for the very first definition of lattice ordered algebras in general is very hard to pinpoint, but originated at around the same time as the previous three references. Indeed, in his review [44] of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts, Frink observed that a general study of lattice ordered rings seems to be needed to study what are now called averaging or Reynolds operators. A call for lattice ordered rings also had gone out by Birkhoff himself in the form of a listed problem at the end of his seminal 1942 paper [20] on lattice ordered groups. Thus lattice ordered algebras and !-algebras seem to have multiple origins, including a study of averaging operators, which themselves sprang forth from problems in fluid mechanics. An appearance at about the same time of !-rings and !-algebras has not resulted in an historical development on complete common ground for these objects. This is not unlike the development of lattice ordered groups versus the development of Riesz spaces. Where the latter have attracted attention from researchers in analysis, the former have been more widely investigated by algebraists. A similar divided attention from analysis versus algebra seems to underlie the connected but somewhat separate tracks of lattice ordered rings versus lattice ordered algebras. Though this separation of tracks is to some extent unavoidable, where each track does have ground that is truly its own, some overlap in results does exist, resulting in difficulties making accurate literature attributions in a survey like ours. We are grateful to two referees for pointing at some references that were missing in our manuscript, though we take full responsibility for possible remaining omissions in the reference list. This survey places itself almost completely on the track of lattice ordered algebras and our only apology for not linking algebra facts in a systematic way to ring results is that all three of us authors were trained as analysts. There is a natural back and forth between the two theories, in one direction by forgetting some of the structure, and in the other by finding, so to speak adjointly, an enveloping algebra. A nice survey on f-rings was written by Henriksen in 1995 (see [50]), to which we refer the interested reader for linkage to some of what follows in this survey. Historically, a lot of the credit for a revival of the theory of !-algebras points to the highly motivating Arkansas Lecture Notes by Luxemburg [67] and the 1982 Ph.D. thesis of de Pagter [76], who systematically explored both the existing literature as well as new directions. Another impetus to research in the area of !-algebras derived from the desire of Zaanen, who in the late seventies started to develop a program to prove many of the elementary results in the theory of Riesz spaces without using representation theorems for vector lattices. This desire is directly linked with a preference not to use the Axiom of Choice unnecessarily. The present survey is intended as an update to the one by Huijsmans [57]. We hasten to point out that we do not intend this survey to replace the one by Huijsmans, but rather that we think of it as augmenting part of it. Since [57] appeared, much progress has been made and several of the problems explicitly phrased in


LATTICE ORDERED ALGEBRAS

101

[57] have been solved. At the same time, some topics like ideal theory, connections between Riesz homomorphisms and algebra homomorphisms, and representations of !-algebras were absent in [57]. Thus an update as well as a supplement was needed. However, some sections of [57] receive no attention at all in this survey. We have not included important topics like the role of !-algebras in positive operator theory (e.g., we do not even include results on the previously mentioned averaging operators) and probability theory. We are rather focused on placing this update, as much as possible, in the setting of spaces of functions, hoping to interest as large an audience as possible via this approach. Moreover, we feel that the great source of inspiration was and continues to be the beautiful book by Gillman and Jerison [47], which certainly inspired and continues to inspire a large part of the research in lattice ordered algebras and rings. Last but not least we focus on what could be called distortions of !-algebras, in particular on Archimedean almost !algebras and d-algebras. Though these distortions have the potential to be seen as aberrations by some, we believe they point the way to techniques that are needed for the broader theory of lattice ordered algebras, as well as for illumination of various aspects in the theory of !-algebras. Note that the distortions disappear if the lattice ordered algebra under consideration has a multiplicative identity which is a weak order unit. Indeed, such algebras are automatically !-algebras. It should be mentioned that classes of lattice ordered algebras other than the ones that appear in this survey have been studied. Notably, the papers [85], [86], [87], and [88] by Steinberg discuss lattice ordered algebras in which every square is positive, a class of algebras that includes all almost !-algebras. We also pay no attention at all to non-Archimedean lattice ordered algebras. Finally, we point out that several results in this survey rely heavily on the (relative) uniform topology on Riesz spaces. In particular, since in Archimedean Riesz spaces uniform limits are unique, we shall include the 'Archimedean' property in the definition of uniformly complete Riesz spaces. A complete investigation of that topology can be found in Sections 16 and 63 of [69]. For terminology and concepts not explained or proved in this survey we refer the reader to the standards books [2], [4 7], [69], [72], [92] and [93]. 2. Definitions and elementary properties A (real) Riesz space A is called a lattice ordered algebra (briefly, an £-algebra) if A also is an algebra and the positive cone

A+

= {f

E

A: f?: 0}

is closed under multiplication, that is, if j,g?: 0 then fg?: 0 (equivalently, if

lfgl::; lfllgl

for all j,g E A).

We make the following blanket assumption: all Riesz spaces under consideration in this paper are assumed to be Archimedean (however, the latter blanket assumption has not stopped us to explicitly add the word Archimedean to the list of conditions in various results below). After Birkhoff and Pierce (see [22, p. 55]), we define an £-algebra A to be an !-algebra if for every j, g E A, the condition

f 1\g

= 0 implies (fh) 1\g = (hf) 1\g = 0 for

all hE A+


102


holds. We call the £-algebra A an almost !-algebra after Birkhoff in [21, Section 6] if

f

Ag

= 0 in A implies f g = 0.

An £-algebra A for which fAg= 0 in A and hE A+ imply (jh) A (gh) = (hf) A (hg) = 0 is called ad-algebra. The notion of d-algebra goes back to Kudlacek in [65]. Our focus in this survey on £-algebras is almost exclusively on f -algebras, almost falgebras and d-algebras. In this paragraph, we recall some properties of !-algebras. Using the Axiom of Choice, Birkhoff and Pierce in [22, Theorem 13] proved that any !-algebra is commutative. A constructive proof of this fact, due to Zaanen, can be found in [61, Theorem 2.1] or [92, Theorem 140.10]. All squares in an !-algebra are positive. Also, lfgl = 111191 for all j, gin an !-algebra A. The multiplication by an element in the !-algebra A is order continuous, i.e., if inf {iT : T} = 0 in A then inf {gfT : T} = 0 for all g E A+. Phrased more generally, the multiplication rrf by an element f E A (rr f (g) = f g for all g E A) is an orthomorphism and all orthomorphisms are order continuous. Recall that an orthomorphism on a Riesz space L is an order bounded linear operator rr such that lrr (f) I A 191 = 0 whenever If I A 191 = 0 in L (the reader is referred to [2] or [92] for elementary properties of orthomorphisms). There is another important relationship between orthomorphisms and !-algebras, which we mention next. Indeed, let Orth (L) be the set of all orthomorphisms on a Riesz space L. Under the operations and the ordering inherited from Lb (L), the ordered algebra of all order bounded operators on L, and under composition as multiplication, Orth (L) is an Archimedean !-algebra with the identity map h on L as unit element. The details of the facts recalled above can all be found in [2],

[76] or [92].

Next we present some properties of almost !-algebras. Almost !-algebras, like !-algebras, are commutative too. The latter fundamental property was first established by Scheffold in [80, Theorem 2.1] for almost !-algebras that are Banach lattices. Using both Scheffold's result and the Axiom of Choice, Basly and Triki were the first to prove commutativity for arbitrary almost !-algebras [10, Thorme 1.1]. The first proof of the commutativity for almost !-algebras within ZermeloFraenkel set theory was given by Bernau and Huijsmans in their paper [13, Theorem 2.15]. Recently, a shorter constructive proof was published in [34, Corollary 3] by Buskes and van Rooij. Another property of !-algebras holds for almost !-algebras, namely the positivity of squares. Also, if A is an almost !-algebra then P = 1!1 2 for all f E A. However, contrary to the order continuity of the multiplication in!algebras, the multiplication by a fixed element in an almost !-algebra is not always order continuous as is shown in the following example. EXAMPLE 2.1. Write A= C ([0, 1]), the vector space of all real-valued continuous functions on [0, 1] . With respect to the pointwise ordering (i.e., f ~ g in A if an only iff (x) ~ g (x) for all x E [0, 1] ), A is an Archimedean Riesz space. Define a multiplication • in A by

f(x)g(x) (f • g) (x) = { f (1/2) g (1/2)

(0

~X~ 1/2); ~X~ 1)

(1/2



103

for all f,g EA. Then A is an almost !-algebra with respect to the multiplication •. For every natural number n ~ 1, define the function fn E A by 1

fn (x)

=

{

n/2- nx ~n/2 + n.T

(0:::; x:::; 1/2- 1/n); (1/2- 1/n :::; x :::; 1/2); (1/2:::; x:::; 1/2 + 1/n); (1/2 + 1/n:::; x:::; 1).

Then sup {fn : n = 1, 2, ... } exists in A and equals the function e defined bye ( x) for all x E [0, 1]. On the other hand, fn (x) (e • fn) (x) = { fn (1/2)

=

1

(0 :::; X :::; 1/2); (1/2:s;x:::;1)

for all n E {1, 2, ... },and clearly the set {e • fn: n = 1, 2, ... } does not have a supremum in A. We conclude that • is not order continuous. For more information about elementary theory of almost !-algebras, the reader is encouraged to consult [13], [23], [34], and [35]. At this point, we turn our attention to some properties of d-algebras. It follows directly from the definition of d-algebras that an £-algebra A is a d-algebra if and only if the multiplication map induced by any fixed element in A+ is a Riesz (or lattice) homomorphism. It follows that a necessary and sufficient condition for an £-algebra A to be a d-algebra is that the identity lfgl = lfllgl holds for all j, g in A. Contrary to (almost) !-algebras, d-algebras need not be commutative nor have positive squares. Next we give an example of a non-commutative d-algebra in which not all squares are positive. ExAMPLE

2.2. Let in this example A be the algebra of real (2

x

2)-matrices of

the form

with the usual addition, scalar multiplication, matrix product and partial ordering. It is not hard to see that A is an Archimedean d-algebra. But A is not commutative and not all squares in A are positive. Indeed, if p= (

~ ~

)

and

q= (

pq

q

and

qp

~ ~

)

then =

=

0.

Moreover, the square -1 ) 0 is not positive.

As for almost !-algebras, multiplication by a fixed element in a d-algebra is, in general, not order continuous. Point in case is the almost !-algebra that we considered in Example 2.1, which also is a d-algebra. Our main reference about d-algebras is [13]. In what follows, we look at some of the connections between the three kinds of £-algebras that we consider in this paper. It is immediate that any !-algebra is both, an almost !-algebra and a d-algebra. Almost !-algebras need not be d-algebras as we see in the next example.


104


EXAMPLE 2,3. Take A as in Example 2.1, and let e E A be the function

e(x) = {

1/2- X 1/2

X-

(0 ~X~ 1/2); (1/2 ~X~ 1).

For f, g E A, define

(f • g) (x)

={

l~:x

f(s)g(s)ds (0

~x~

1/2);

(1/2~x~1).

e(x)f(x)g(x)

Then A is an almost f -algebra under the multiplication •· How ever, A is not a d-algebra. Indeed, let e, f in A be defined by e (x) = 1 and f (x) = {

Observe that

If • gl

=

If • el (0) = 0

If I • lgl

and

for all x E [0, 1]

-4x + 1 4x-3

(0 ~ x ~ 1/2);

!

(1/2~x~1).

(lfl• e) (0) =

1

1/2

lf(s)l ds-=/= 0.

Thus the property

fails in A.

Since every almost !-algebra is commutative and has positive squares, Example 2.2 shows that d-algebras need not be almost !-algebras. However, if ad-algebra A is commutative or has positive squares then A is automatically an almost !-algebra [22, p. 60]. Summarizing part of the relations, we have the following diagram !-algebra=? commutative d-algebra =? almost !-algebra. For more detail, see [22], [13] and [57]. The next lines deal with nilpotent element in £-algebras. The set of all nilpotent elements in the £-algebra A is denoted by N (A). In other words,

N (A)= {f E A:

r

= 0 for some n = 1, 2, ... }.

Given a natural number p, we define NP (A)= {f E A:

fP =

0}.

The £-algebra A is said to be semiprime if 0 is the only nilpotent element in A, that is, if N (A) = {0}. If A is an !-algebra then the following equalities hold

N (A)= N2 (A)= {f

E:O

A: fg

=

0 for all g E A}.

(see [76, Proposition 10.2] or [92, Theorem 142.5]). If A is an almost !-algebra then

N (A) = N3 (A) = {f

E A: fg 2

= 0 for all

g E A}

= {f E A: fgh = 0 for all g, hE A}

(see [51, Theorem 3.11]) and, as for !-algebras, N2 (A)= {f E A: fg

= 0 for all g

E A}

(see [23, Lemma 5.3]). If A is ad-algebra then

N (A)= N 3 (A)={! E A: gfh = 0 for all g, hE A} (see [13, Theorem 5.5] or [29, Theorem 5]). However, and contrary to what we documented for !-algebras and almost !-algebras, in d-algebras the equality



105

N 2 (A) = {f E A : f g = 0 for all g E A} does not necessarily hold as is illustrated by the following example. EXAMPLE 2.4. Take A as in Example 2.1 and define the multiplication • in A by (f•g)(x)=f(O)g(1) forallj,gEA. Let f E A be the function defined by f (x) = 1- x for all x E [0, 1]. Clearly, f • f = 0 but f • e =J 0 where e E A is defined by e ( x) = 1 for all x E [0, 1].

Finally, note that any !-algebra with multiplicative identity is semiprime and any semiprime almost !-algebra or semiprime d-algebra is automatically an !algebra (see Section 1 in [13]). As a final comment we remark that an [-algebra which has positive squares and has a multiplicative identity need not be an !algebra. 3. £-algebra multiplications in C (X) Let C (X) be the set of all real-valued continuous functions on a compact Hausdorff topological space X. Under pointwise addition and scalar multiplication, C (X) is a real vector space. Moreover, C (X) is an Archimedean Riesz space with respect to the pointwise ordering (i.e., f::; gin C (X) if and only iff (x) ::; g (x) for all x E X). By defining the multiplication in C (X) pointwise as well (i.e., (!g) (x) = f (x) g (x) for all j, g E C (X) and all x EX), the space C (X) is easily seen to have the structure of an !-algebra with e as unit element, where e (x) = 1 for all x EX. Now consider another associative multiplication • in C (X). The main topic of this section is to produce necessary and sufficient conditions for C (X) to be an !-algebra (respectively, an almost !-algebra, ad-algebra) with respect to this new multiplication •. The first theorem in this direction goes back to Conrad (see [38, Theorem 2.2]), who obtained the following. THEOREM 3.1. Let • be an associative multiplication inC (X). Then C (X) is an f -algebra with respect to • if and only if there exists a positive function w E C (X) such that (f•g)(x) =w(x)f(x)g(x) for all j,g E C(X) and all x EX. In fact, Conrad established the theorem above for any Archimedean f-ring with unit element. The representation formula given in Theorem 3.1 above was obtained in an alternative way by Scheffold in [80, Korollar 1.4]. While Conrad's proof is purely algebraic and order theoretic, the proof presented by Scheffold relies on analytic tools like the Riesz representation theorem. With the same analytic tools Scheffold also obtained the following representation theorem for almost !-algebra multiplications inC (X) (see [80, Theorem 1.2]). THEOREM 3.2. Let • be an associative multiplication inC (X). Then C (X) is an almost f -algebra with respect to • if and only if there exists a family (f.-lx : x E X) of positive measures such that (f • g) (X)

= [ J (S) g (S) df.Lx ( S)

for all j,g E C(X) and all x EX.


106


Since every commutative d-algebra is an almost !-algebra, the previous theorem remains valid for commutative d-algebra multiplications inC (X) as well. Recently in [25, Corollary 3.2], Boulabiar proved the following representation formula for any (not necessarily commutative) d-algebra multiplication inC (X). THEOREM 3.3. Let • be an associative multiplication inC (X). Then C (X) is a d-algebra with respect to • if and only if there exist

(i) a positive function w E C (X), and (ii) functions h,k: X---+ X (continuous on coz(w) = {x EX: w(x) =f. 0}) such that (f • g) (x) = w (x) f (h (x)) g (k (x)) for all j, g E C (X) and all x E X. Notice that if C (X) is a d-algebra with respect to the multiplication • then • is commutative if and only if the functions h and k coincide on coz (w) (where h, k and w are as in Theorem 3.3). The latter observation yields, in addition to the formula cited in Theorem 3.2 above, another representation for commutative d-algebra multiplications on C (X). More abstract versions of the results above will be given in Section 8 below.

4. Multiplication by an element as an operator Let A be an £-algebra and recall that .Cb (A) denotes the ordered algebra of all order bounded operators on A. For every f E A, we define the map 1rf on A by 1r f (g) = f g for all g E A. Clearly, 1r f is an order bounded operator on A for all f E A. The map p : A ---+ .Cb (A) defined by p (f) = 1rf for all f E A is obviously an algebra homomorphism, that is, p (fg) = p (f) p (g) for all j, g E A. Hence the range p (A) of pis a subalgebra of .Cb (A). In this section, we will see that if A is an almost !-algebra then p (A) can canonically be equipped with an ordering, under which p (A) is an Archimedean !-algebra. A corresponding result will also be given for commutative d-algebras and !-algebras. Let A be an almost !-algebra. Since N 2 (A)= {f E A: .fg = 0 for all g E A},

= 1r9 if and only if .f - g E N 2 (A). This allows us to define an ordering on p (A) by putting

1r f

(0) The ordering defined by (0) coincides with the ordering inherited from .Cb (A), namely, 1rf is positive with respect to ( 0) if and only if 1rf is a positive operator on A. Under the usual addition and composition of operators, and with the ordering defined by (0), p (A) is an Archimedean ordered subalgebra of .Cb (A). In fact, we have the following theorem (see Theorem 4.2 and Theorem 4.4 in [23]). THEOREM 4.1. Let A be an Archimedean almost !-algebra. Then p (A) is an A rchimedean .f -algebra with respect to the addition and composition of operators, and the ordering inherited .from Lb (A). The lattice operations in p (A) are given by

1fJ V1r9

= 1fJv 9 ,

1fJ

/\1r 9

= 1fJ 11 g for all j,g

EA.

In particular, (7rJ)+=1fJ+,

(7rJ)-=7r 1 -,

17rtl=7riJI

.forall.fEA.



107

In other words, p defines a surjective Riesz homomorphism from A onto p (A). Theorem 4.1 of course holds in commutative d-algebras. Moreover, let A be a commutative d-algebra. For f E A and g E A+ the equalities l1rtl (g)= 1r1t1 (g)= III g =sup {lfllhl: lhl :S g} =sup {If hi :I hi :S g} =sup {11ft (h)l : Ihi :S g} imply that 1r1 has an absolute value in Lb (A) , which coincides with its absolute value in p (A). We collect the latter observations for commutative d-algebras. THEOREM 4.2. Let A be an Archimedean commutative d-algebra. Then p (A) is an Archimedean !-algebra when equipped with the addition and composition of operators, and the ordering inherited from Lb (A). Moreover, the absolute value 1rl/l of 1r1 in p (A) coincides with the absolute value of 1r1 in Lb (A) for all f E A, that 7S,

l1rtl (g)= 1rl/l (g)= sup{l7rJ (h)l: lhl :S g}

for all g E A+.

We obtain the !-algebra case as a corollary. CoROLLARY 4.3. Let A be an Archimedean !-algebra. Then p (A) is an fsubalgebra of the Archimedean f -algebra Orth (A) of all orthomorphisms of A.

The fact that the range of pin Corollary 4.3 is an !-algebra was first proved in [22, Corollary 3, p. 57] by Birkhoff and Pierce, while the fact that Orth (A) itself is an !-algebra has been proved in [18] by Bigard and Keimel and in [39] by Conrad and Diem. This topic was also discussed in great detail by de Pagter in his thesis [76, Proposition 12.1]. Note that if A is an !-algebra then A is semiprime if and only if p is one-to-one as a map from A into Orth (A). In this case, A and p (A) are isomorphic as !-algebras. Also, if A is an !-algebra then A has a multiplicative identity if and only if the map p is one-to-one and onto as a map A ---+ Orth (A), and consequently A and Orth (A) are isomorphic as !-algebras.

5. Uniform completion and Dedekind completion Let A be an Archimedean £-algebra. The closure Aru of A in its Dedekind completion A 5 with respect to the uniform topology is a uniformly complete Riesz space. Using Quinn's Definition 2.12 in [79], Aru is the uniform completion of A. The following theorem was obtained by Triki in [91]. THEOREM 5.1. Let A be an Archimedean £-algebra (respectively, almost !algebra, d-algebra, f -algebra). Then the multiplication in A extends uniquely to a multiplication in Aru such that Aru is a uniformly complete £-algebra (respectively, almost f -algebra, d-algebra, f -algebra) with respect to this extended multiplication. Moreover, if A is semiprime (respectively, has a unit element e) then Aru is semiprime (respectively, has e as unit element).

We now turn our attention to the Dedekind completion of the £-algebra A. Johnson in his paper [64] proved that if A is an !-algebra (or even an Archimedean f-ring), then the multiplication in A extends uniquely to an !-algebra multiplication in A 0 . The uniqueness of such an extended multiplication in A 0 of course arises from the order continuity of the multiplication in the !-algebra A. Alternative proofs of this extension can be found in [76, pp. 66-67] and [59, p. 166].


108


THEOREM 5.2. Let A be an Archimedean !-algebra. Then the multiplication in A extends uniquely to a multiplication in A 6 such that A 6 is a Dedekind complete!algebra with respect to this extended multiplication. Furthermore, if A is semiprime (respectively, A has a unit element e) then A 6 is semiprime (respectively, has e as unit element).

The corresponding results for £-algebras in general, or almost /-algebras and d-algebras in particular, is much harder because of the absence of order continuity of the multiplication. Nonetheless, extensions of the multiplication to the Dedekind completion often exist, though such extensions are no longer necessarily unique. For almost /-algebras Buskes and van Rooij proved the following (see [35, Theorem 10]). THEOREM 5.3. Let A be an Archimedean almost !-algebra. Then the multiplication in A extends to a multiplication in A 6 such that A 6 is a Dedekind complete almost f -algebra with respect to that extended multiplication.

Using the previous result as a starting point, Boulabiar and Chil in [28, Corollary 3] proved that from amongst the extensions provided, A 6 can be equipped with a commutative d-algebra multiplication whenever A is a commutative d-algebra. Then in [37, Theorem 7], Chil was able to drop the commutativity condition and prove the following theorem. THEOREM 5.4. Let A be an Archimedean d-algebra. Then the multiplication in A extends to a multiplication in A 6 such that A 6 is a Dedekind complete d-algebra with respect to that extended multiplication.

In summary, all but one of the problems concerning Dedekind completions that Huijsmans raised in his survey paper [57] have now been solved. The remaining problem, though admittedly outside the scope of this survey, is the following. PROBLEM 5.5. Let A be an Archimedean £-algebra. Does the multiplication in A extend to a multiplication in A 6 so that A 6 is a Dedekind complete £-algebra?

6. Powers in £-algebras Let A be a uniformly complete £-algebra and let P E JR+ [X1 , ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. In their paper [16], Beukers and Huijsmans considered the following problem: does there exist in A a 'p-th root' of P (!1, ... , fn) for fi, ... , fn in A+? They gave an affirmative answer in the case where A is a semiprime !-algebra. More precisely, they proved the following theorem (see [16, Theorem 5]). THEOREM 6.1. Let A be a uniformly complete semiprime !-algebra and let P E JR+ [X1 , ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. Then for every fi, ... , fn E A+ there exists a unique f E A+ such that fP = p (fi, ... , fn)·

As a consequence, one has the following corollary (see Corollary 6 in [16]). CoROLLARY 6.2. Let A be a uniformly complete !-algebra with unit element and p E {1, 2, ... }. Then for each f E A+, there exists a unique g E A+ such that

gP

=f.



109

Note that the previous result was first proved for p = 2 in [17, Theorem 4.2 and Cororllary 4.3] by Beukers, Huijsmans and de Pagter. Also, it should be noted that Theorem 6.1 above is proved alternatively by Buskes, de Pagter, and van Rooij in [31, Corollary 4.11], a paper that deals with a more general functional calculus on Riesz spaces and !-algebras to which we will return in the next section. The problem corresponding to Theorem 6.1 for almost !-algebras was considered by Boulabiar and follows next (see Theorem 3 in [24]). THEOREM 6.3. Let A be a uniformly complete almost f -algebra and let P E JR+ [X 1 , ... , Xn] be a homogeneous polynomial of degree a natural number p. Then for every h, ... , f n E A+ there exists a (not necessarily unique) f E A+ such that JP = P (JI, ... , fn)·

Observe that, where roots are unique in semiprime !-algebras, this is no longer always the case for almost !-algebras. We illustrate this with an example. EXAMPLE 6.4. Let A = C ( [-1, 1]) be the uniformly complete Riesz space of all real-valued continuous functions on [-1, 1] and define w E A by

~x

w(x)={

(-1:::;x:::;O); (O:::;x:::;1).

For every j, g E A, we put

(f • g) (x) =

{

(-1:::;x:::;O);

w(x)f(x)g(x) 0

[ /(s)g(s)ds

(O:::;x:::;1).

Then A is an almost f -algebra with respect to the multiplication •. h, g, a, (3 E A defined by

g(x)=lxl

h (x)

=

Consider

exp (x) ,

and

a (x) for all x Then

E

=

jx 2

+ exp (2x)

[-1, 1], where

X[o,l]

, and (3 (x) (x)

=

1 if x

E

+ )x 2 + exp (2x)

= X-X[O,l]

(x)

[0, 1] and

X[O,l]

a • a = (3 • (3 = g • g

(x)

=

0 if x

E

[-1, 0).

+ h • h.

At this point, we define for each non zero natural number p, Ap

= {h · · · fp: JI, ... , JP

E

A}.

In what follows, we will investigate the order structure as well as the algebra structure of Ap (since A 1 = A, we suppose that p ;::: 2). The sets A 2 and A 3 were first considered in [35] by Buskes and van Rooij and then in great detail by Boulabiar in [24] from which we summarize the results in the following theorem (see Theorems 4, 5, and 6 in [24]). THEOREM 6.5. Let A be a uniformly complete almost !-algebra and let p ;::: 3 be a natural number. Then Ap is a uniformly complete semiprime f -algebra under the ordering and multiplication inherited from A. The positive cone At of AP is defined by


110


The lattice operations 1\p and VP in Ap are given by

fP 1\p gP = (f 1\ g)P and the absolute value

l-IP

and

fP

Vp gP

= (f

V g)P for all 0::; f, g E A,

in Ap is defined by

IJPIP = IJIP

for all f EA.

Contrary to Ap (p 2': 3), A 2 need not be a Riesz space under the ordering inherited from A as is proved by the next example. EXAMPLE 6.6. Consider A= C ([0, 1]) with the pointwise addition, scalar multiplication and partial ordering. For J, g E A, define

lx-1/2

(f • g) (x) = {

0 f(s)g(s)ds

0

(0 ::;

X ::;

1/2);

(1/2..IA for some real number .X} is the centre of A (i.e., the principal order ideal generated by IA in Orth (A)). We denote, after Triki (see [90]), the set of all bounded elements in A by Ab. Note that rather than Ab, Henriksen and Johnson use A* in [53] after a similar usage in Gillman and Jerison's book. Clearly, Ab is an f-subalgebra of A. The !-algebra A is said to be bounded if A= Ab, that is, if p (A) is a subset of Z (A). In particular, a unital !-algebra A is bounded if and only if its multiplicative identity is also a strong order unit, and in this situation A and Z (A) are isomorphic as !-algebras. Here we recall that the identity element in an Archimedean unital !-algebra is a weak order unit (see [22, p. 60]). The equivalence of (i) and (ii) in the next theorem is an immediate consequence of the definitions of an order ideal and a bounded Archimedean !-algebra, while for the proof of the rest of this theorem, we refer to Theorem 3 in [11] and Theorem 4 in [12]. THEOREM 13.2. Let A be an Archimedean !-algebra. Then the following are equivalent. (i) Every order ideal is a ring ideal. (ii) A is bounded.



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If A is in addition uniformly complete then (i) above is equivalent to each of the following. (iii) Every maximal modular ring ideal in A is uniformly closed. (iv) Every maximal modular ring ideal is the kernel of a Riesz and algebra homomorphism from A onto JR. (v) Every maximal modular ring ideal is a maximal order ideal. We turn our attention to semiprime !-algebras. Let L be a Riesz space. A norm 11.11 on a Lis called a Riesz norm if 11!11 ~ llgll whenever lfl ~ lgl in L. If a Riesz norm on L exists then L is said to be a normed Riesz space. If the normed Riesz space L is a Banach space as well, we say that L is a Banach lattice. We call the Riesz norm 11.11 on L an M-norm if II! V gil =max {11!11, llgll} for all f, gEL. An !vi-space is an M-normed Banach lattice. We can now state the following theorem (see [11, Theorem 5] and [12, Corollary 5]). THEOREM 13.3. Let A be an Archimedean semiprime f -algebra. The following are equivalent. (i) Every order ideal in A is a ring ideal in A. (ii) A is a isomorphic as an f -algebra to a subalgebra of Z (A). (iii) There exists an M -norm in A. (iv) There exists a Riesz norm in A. If, in addition, A is uniformly complete then each of (i), (ii), (iii) and (iv) above is equivalent to (v) Every maximal modular ring ideal in A is uniformly closed. We note that Problem (P1) was extensively studied by Henriksen, Larson and Smith [52] in the context off-rings. We move on to discuss Problem (P2). To this end, we need the notion of a normal Riesz space. The Riesz space L is said to be normal if L = {!+} d +{!-} d for all f E L, where {j+}d ={gEL: lgl/\f+ = 0} and {f_}d ={gEL: lgl/\f- = 0}. For a completely regular topological space X, the Riesz space C (X) is normal if and only if the sets P(f) = {x EX: f(x) > 0} and N(f) = {x EX: f(x) < 0} are completely separated for every f E C (X). For spaces of the type C (X), Problem (P2) has the following solution (see Theorem 14.24 in [47]). THEOREM 13.4. Let X be a completely regular (Hausdorff) topological space. Then the following are equivalent. (i) Every ring ideal in C (X) is an order ideal. (ii) Every finitely generated ring ideal in C (X) is a principal ring ideal (i.e., X is an F -space). (iii) C (X) is normal. Huijsmans and de Pagter considered (P2) in !-algebras. They only considered the unital case and their central result is the following generalization of Theorem 13.4 above (see [58, Theorem 6.6] for the proof). THEOREM 13.5. Consider the following conditions for an Archimedean !-algebra A with unit element. (i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal.


124


(iii) A is normal. Then (i) ::::} (ii) ::::} (iii). If, in addition A is uniformly complete then (i) (ii) (iii). Next we focus on the non unital case, which was considered by Triki in [90]. First we define the notion of a stable !-algebra. The !-algebra A is said to be stable if 1r (f) E (f) (where (f) is the principal ring ideal generated by f in A) for all 1r E Z (A). It is clear that A is stable if and only if 1r (I) C I for all ring ideals I in A and all 1r E Z (A). We now are in a position to present the result corresponding to Theorem 13.5 for semiprime !-algebras (see [90, Theorem 5.4]). THEOREM 13.6. Consider the following conditions for an Archimedean semiprime f -algebra A. (i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal. (iii) A is stable and normal, and {j+}d or {f_}d is a modular ring ideal for all f EA. Then (i) ::::} (ii) ::::} (iii). Furthermore, if A is in addition uniformly complete then (i) {:} (ii) {:} (iii). We proceed to the non semiprime case (see Theorem 5.5 in [90]). THEOREM 13.7. Let A be a non semiprime Archimedean !-algebra. Then the following are equivalent. (i) Every ring ideal is an order ideal. (ii) There exist a semiprime !-algebra B such that (a) every ring ideal in B is an order ideal, and (b) { b} d is a modular ring ideal in B for every b E B, so that A is isomorphic to the f -algebra B x JR. endowed with the multiplication defined by (!,a) (g,{3) = (fg,O) for all J,g E B;a,{3 E R Finally, note that the only £-algebras that we have considered in this section are !-algebras. Indeed, a Problem (P1) for more general £-algebras is futile since an £-algebra in which every order ideal is a ring ideal automatically is an !-algebra (see Page 144 in the classical book [42] or Proposition 1 in [12]). The situation for Problem (P2) is less clear. Indeed, in matrix algebras and algebras of formal power series in one variable, when ordered coordinatewise, every ring ideal is an order ideal. Thus we phrase Problem (P2) for lattice ordered algebras in general. PROBLEM 13.8. Study Problem (P2) for Archimedean £-algebras other than f -algebras.

14. Representation of /-algebras Let A be an !-algebra and recall that Ab denotes the f-subalgebra of all bounded elements in A (see Section 13 above). Several properties are satisfied by A if and only if they are satisfied by Ab and vice versa (see Sections 3, 4 and 5 in [90]). Thus we can study some aspects of A via an investigation of Ab. For instance, if we assume that every ring ideal in A is an order ideal then A b has the same property and the converse holds if A in addition is uniformly complete (see [84] by Steinberg). In particular, if every ring ideal in A is an order ideal then order


125


ideals and ring ideals coincide in Ab (see Theorem 13.2). It turns out that under the latter hypothesis, A b has a nice representation as a space of functions. This kind of a representation then is precisely the topic of this section. First assume that A has an identity element (i.e., A is a -algebra in the Henriksen-Johnson terminology of [53]). The set of all maximal £-ideals in A will be denoted by M (A). For any subset D of M (A), the kernel k (D) of D is k (D)=

n {M: ME D}

(where it is understood that k (0) = A). The hull h (I) of an £-ideal I in A is

h (I)= {ME M (A): I

c

M}.

The subset D of M (A) is said to be closed if D = h (k (D)). One thus defines the hull-kernel (or Stone) topology on M (A). It turns out that M (A) with respect to this hull-kernel topology is a compact Hausdorff space (see [53, Theorem 2.3]). Suppose at this point that A is in addition uniformly complete (instead of 'uniformly complete', Henriksen and Johnson in [53] use 'uniformly closed') and denote the identity element of A by e. Since A is unital, Ab is precisely the principal order ideal generated by e. Also recall that A and Orth (A) are isomorphic as !-algebra under the given condition. As usual, the !-algebra of all real-valued continuous functions on the compact Hausdorff space M (A) is denoted by C (M (A)). Henriksen and Johnson in [53, 3.2, p. 84] proved, using the Stone-Weierstrass theorem, the following variant of Stone's representation theorem. THEOREM 14.1. Let A be a uniformly complete f -algebra with identity element e. Then Ab and C (M (A)) are isomorphic as !-algebras. In particular, if e is a strong order unit in A then A and C (M (A)) are isomorphic as f -algebras.

The latter result can of course also be obtained via Kakutani's representation theorem. Indeed, being the principal order ideal in A generated by e, Ab is a uniformly complete Riesz space with e as a strong order unit. It follows that Ab is an M-space with respect to theM-norm ll·lle defined by

llflle =

inf {A> 0:

lfl

~

Ae}

for all

f

EA

(see [72, Proposition 1.2.13]). Kakutani's representation theorem (see, for instance, [72, Theorem 2.1.3]) guarantees the existence of a compact Hausdorff space 0

so that Ab and C (0) are isomorphic as !-algebras and isometric as M-spaces. From an investigation of the cited proof of Kakutani's theorem, we see that n is the set of all algebra homomorphisms from Ab onto JR, or, equivalently, the set of all real valued Riesz homomorphisms that send e to 1. With respect to the weak topology a ((Ab)~ ,Ab), where (Ab)~ is the order dual of Ab, the set n is a a ((A b)~, Ab)-compact Hausdorff space. In view of Theorem 13.2, we observe that n is homeomorphic to M (Ab) and then to M (A) since M (Ab) and M (A) are also homeomorphic (see [53, Corollary 2.8]). We thus recover Theorem 13.1 above. The reader should also compare the above with Gelfand theory for Banach algebras

[94].

We proceed to representation theorems for non unital !-algebras. For the terminology and notations concerning the topological spaces under consideration, we refer the reader to the book [47] by Gillman and Jerison. The following two theorems deal with semiprime !-algebras (see Theorems 7.2 and 7.9 in [90]).


126


THEOREM 14.2. Let A be a uniformly complete semiprime !-algebra with a weak order unit. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists an almost compact F-space D such that Ab and C 0 (D) are isomorphic as f -algebras.

For almost compact F-spaces, we refer to [47, 6J]. THEOREM 14.3. Let A be a semiprime Dedekind a-complete !-algebra. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exist a basically disconnected locally compact Hausdorff space D1 and a basically disconnected almost compact Hausdorff space D2 so that Ab is isomorphic as an !-algebra to one of the algebras

CK (Dl), Co (D2) or CK (Dl) EB Co (D2). Our next representation theorem represents non semiprime !-algebras (see [90, Theorem 7.10]). THEOREM 14.4. Let A be a uniformly complete non semiprime !-algebra A. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists a locally compact F'-space D such that Ab is isomorphic as an !-algebra to the Cartesian product Co (D) x IR, where the multiplication in the !-algebra C 0 (D) x lR is defined by

(!,A) (g, p,)

= (fg, 0)

for all j, g E Co (D) and A, p, E R

Note that the topological spaces D in Theorem 14.2 and in Theorem 14.4 are constructed in the same way as follows: for the uniformly complete !-algebra A (semiprime or not), the centre Z (A) is a uniformly complete !-algebra and its multiplicative identity fA is a strong order unit. According to Theorem 14.1, Z (A) is isomorphic as an !-algebra to the !-algebra C (M (Z (A))) of all real-valued continuous functions on the compact Hausdorff topological space M (Z (A)) (the set of all maximal €-ideals in Z (A)). It turns out that D = U {coz (f):

f

E

C (M (Z (A)))}.

In a similar way, the topological spaces D 1 and D2 in Theorem 14.3 are constructed from a certain uniformly complete f-subalgebra of A (for more information, we refer to [90]). We end this section with two comments. Observe that representation theorems listed in this section apply to Ab and not to the whole !-algebra A. What one can say about A itself is the following. If A is unital then A can be embedded as an !-algebra into an algebra of extended functions on M (A) [53, Theorem 2.3], and if A is semiprime then A can be considered as an f-subalgebra of M (Orth (A)) (recall that if A is semiprime then A can be considered as an f-subalgebra of Orth (A)). In addition, the representation theorems that we presented in this section are not necessarily excessively restrictive due to the previously mentioned fact about transference of various properties of Ab to A. Our final comment about representing lattice ordered algebras deals with a matter of set theory. Zaanen started an ambitious program at around 1980, intending to prove all available material in vector lattices in as far as possible in an



127

elementary way, i.e., without using representation theorems. A commonly quoted reason to adhere to Zaanen's program is the need to not use the Axiom of Choice unnecessarily. It is therefore interesting to know that one can avoid the Axiom of Choice and still use representation theorems, as long as the constructs that one has in mind depend on say countably many elements of a Riesz space. For !-algebras one of the most useful theorems in that direction is the following (combine Theorem 2.2 in [33] and Corollary 2. 7 in [36]). THEOREM 14.5. Let A be a semiprime !-algebra. If D is a countable subset of A then the f -subalgebra generated by D in A can be represented within ZermaloFraenkel set theory as an f -subalgebra of the space of continuous functions on a metric space.

It should be noted at that the connection between the Axiom of Choice and representations of !-rings in terms of subdirect products of totally ordered algebras was discussed in the works [42] by Feldman and Henriksen, [67] by Luxemburg, and [9] by Banaschewski. Contrary to the possibility of locally representing countably many elements of any semiprime !-algebra as continuous functions without any choice, the global representation of !-algebras as subdirect products of totally ordered algebras can not be obtained without appealing to some transcendent tool from set theory. The latter kind of representation theorem is important for more than historical reasons. First of all, many researchers define !-algebras as subdirect products of totally ordered algebras. Secondly, Birkhoff and Pierce in their seminal paper [22] observed that the Axiom of Choice seems to be involved if one wishes to obtain (with the definition for !-algebras as in this survey) a representation theorem for !-algebras as a subdirect product of totally ordered algebras, which for convenience we will now name the Birkhoff-Pierce Representation Theorem. The three papers [42], [67], and[9] independently prove that the Boolean Prime Ideal Theorem is both sufficient as well as needed for the Birkhoff-Pierce Representation Theorem. Thus Stone's Representation Theorem for Boolean algebras is constructively equivalent to the Birkhoff-Pierce Representation Theorem for !-algebras. In turn, each of the latter representation theorems is constructively equivalent to the Kakutani Representation Theorem for vector lattices with a strong order unit. In about that same direction, we observe that it is still unknown whether the Boolean Prime Ideal Theorem suffices for that other main representation tool for vector lattices as vector sublattices of extended real valued continuous functions, the so-called Maeda-Ogasawara Representation Theorem (see [32]).

15. Linear biseparating maps on /-algebras A linear map T between two algebras A and B is said to be separating if T (f) T (g) = 0 in B whenever f g = 0 in A. If in addition T is bijective and its inverse T- 1 is separating as well then T is said to be biseparating. Clearly, if T is one-to-one and onto then Tis biseparating if and only if fg = 0 in A{:} T (f) T (g)= 0 in B. If A and Bare assumed to be !-algebras with unit elements, then Tis separating if and only if T is disjointness preserving, that is, If I/\ 191 = 0 in A implies IT (f) lA IT (g) I = 0 in B. This follows directly from the equivalence

fg

= 0 {:} lfl/\ 191 = 0,


128


which holds in any semiprime !-algebra. For the same reason, the linear map T between two !-algebras with multiplicative identity is biseparating if and only if Tis ad-isomorphism (i.e., Tis bijective and both T and T- 1 are disjointness preserving) The reader is encouraged to consult the beautiful memoir [1] by Abramovich and Kitover for the theory of disjointness preserving linear maps on Riesz spaces. The study of when linear biseparating maps on algebras of real or complex valued continuous functions are weighted isomorphisms started in 1990 with the paper [63] by Jarosz and culminated in the work [5] by Araujo, Beckenstein and Narici with the following result. Let C (X) and C (Y) be the algebras of real or complex valued continuous functions on completely regular topological spaces X and Y, respectively. If T is a linear biseparating map then there exist a nonvanishing w E C (Y) and an homeomorphism h from the realcompactification vX of X onto v Y, such that

T (f) (y)

= w (y) f (h (y))

for all

f

E

C (X) and y E Y.

Henriksen and Smith in [55] explored the aforementioned result by Araujo, Beckenstein and Narici in the more general setting of unital !-algebras. They proved that every positive linear biseparating map T between two unital !-algebras A and B closed under inversion is a weighted isomorphism, that is, there exist an invertible w E B and a Riesz isomorphism S : A -+ B which is simultaneously an algebra isomorphism, such that

T (f)= wS (f)

for all

f

EA.

Very recently in [30], Boulabiar, Buskes, and Henriksen extended the latter result to all order bounded linear biseparating maps on arbitrary (not necessarily closed under inversion) unital !-algebras over the reals as well as over the complex numbers (for the theory of complex !-algebras, we refer to [17]). The theorem they obtained is the following. THEOREM 15.1. Let A and B be (real or complex) !-algebras with unit elements, and let T : A -+ B be an order bounded linear biseparating map. Then T is a weighted isomorphism.

In the previously mentioned memoir by Abramovich and Kitover, we find the following theorem. A d-isomorphism between two uniformly complete Riesz spaces A and M is automatically order bounded as soon as every universally CJ-complete projection band in A is essentially one-dimensional (see [1, Corollary 15.3]). This theorem is used in [30], under the same conditions on A, to show that every linear biseparating map between two uniformly complete !-algebras A and B is a weighted isomorphism. We point out that a complex !-algebra is by definition uniformly complete. Thus the phrase 'uniformly complete unital !-algebra' is understood to mean either a uniformly complete unital !-algebra over the reals, or simply a unital !-algebra over the complex numbers. For the proof of the next theorem, see Proposition 5.1 and Theorem 5.2 in [30]. THEOREM 15.2. Let A and B be uniformly complete unital f -algebras and assume that every universally CJ-complete projection band in A is essentially onedimensional. Then every linear biseparating map from A onto B is order bounded and then a weighted isomorphism.

To obtain the result by Araujo, Beckenstein and Narici cited above as a consequence of the preceding theorem, Boulabiar, Buskes and Henriksen proved that



129

every algebra of all scalar-valued continuous functions on a completely regular topological space has the property that every universally a-complete projection band is essentially one-dimensional (see [30, Theorem 5.5]). THEOREM 15.3. Let X be a completely regular topological space X. Then every universally a-complete projection band in the Riesz space C (X) is essentially onedimensional.

Combining Theorems 15.2 and 15.3, we arrive at the next result. COROLLARY 15.4. Let X and Y be completely regular topological spaces. Then every biseparating linear map T : C (X) ___, C (Y) is a weighted isomorphism. In particular, C (X) and C (Y) are isomorphic as !-algebras if and only if there exists a linear biseparating map from C (X) onto C (Y).

It is well known that if X and Y are completely regular topological spaces and Sis an isomorphism from C (X) into C (Y) then there exists an homeomorphism h from vY into vX such that S (!) = f o h, where vX and vY denote the realcompactifications of X and Y, respectively (see Section 10 in [47]). The latter fact, together with Corollary 15.4 above, directly leads to the next corollary, which was proved earlier in an alternative way by Araujo, Beckenstein and Narici in [5, Proposition 3]. COROLLARY 15.5. Let X and Y be completely regular topological. Then for every linear biseparating map T : C (X) ---> C (Y) there exist a non-vanishing function wE C (Y) and an homeomorphism h: vY---> vX such that

T(!)(y)=w(y)f(h(y))

forallfEC(X) andyEY.

It is shown in [47, Theorem 8.3] that two realcompact X andY are homeomorphic if and only if C (X) and C (Y) are isomorphic as !-algebras. Another classical result of rings of continuous functions theory is that if X is a completely regular topological space then C (X) and C (vX) are isomorphic as !-algebras (see Remark 8 (a) in [47]). It follows immediately that if X and Y are two completely regular topological spaces, and C (X) and C (Y) are isomorphic as !-algebras, then vX and vY are homeomorphic. The latter implies that, without further assumptions, the conclusion that vX is homeomorphic to vY in Corollary 15.5 is best possible. Under additional assumptions, however, X and Y may themselves be homeomorphic as is shown in the next Corollary. COROLLARY 15.6. Let X andY be completely regular topological spaces and assume either (i) or (ii) below. (i) X and Y are realcompact. (ii) The points of X, as well as those ofY, are G 0 -points. If there exists a linear biseparating map from C (X) onto C (Y) then X andY are homeomorphic.

Under the condition (i), the result in Corollary 15.6 above follows straightforwardly from Corollary 15.5 while under the condition (ii), it stems directly from [73] by Misra. Finally, it should be noted that the result of Theorem 15.2 above is false if the universally a-complete projection bands in A fail to be essentially one-dimensional. Indeed, Example 7.13 in [1] produces a linear biseparating map Ton the Dedekind


130


complete (and then uniformly complete) unital /-algebra L 0 ([0, 1]) of all equivalence classes of measurable functions on [0, 1], which is not order bounded and thus cannot be a weighted isomorphism. References [1] Abramovich, Y. A. and A. K. Kitover, Inverses of disjointness preserving operators, Memoirs Amer. Math. Soc., 143 (2000), no 679. [2] Aliprantis, C. D. and 0. Burkinshaw, Positive Operators, Academic Press, Orlando, 1985. [3] Amemiya, 1., A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser I, 12 (1953), 111-156. [4] Anderson, F. W., Lattice-ordered rings of quotients, Canad. J. Math., 17 (1965), 434-448. [5] Araujo, J., E. Beckenstein and L. Narici, Biseparating maps and homeomorphic realcompactifications, J. Math. Ana. Appl., 12 (1995), 258-265. [6] Arens, R., Operations induced in function classes, Monatshefte Math., 55 (1951), 1-19. [7] Arens, R., The adjoint of bilinear operation, Proc. A mer. Math. Soc., 2 (1951), 839-848. [8] Arens, R. and J. L. Kelley, Characterizations of the space of continuous functions over a compact Hausdorff space, Trans. A mer. Math. Soc., 62 (1947), 499-502. [9] Banaschewski, B., The Prime Ideal Theorem and representations of /-rings, Algebra Univ., 25 (1988), 384-387. [10] Basly, M. and A. Triki, F F -algebres Achimediennes, Pre print. [11] Basly, M. and A. Triki, /-algebras in which order ideals are ring ideals, Indag. Math. 50 (1988), 231-234. [12] Basly, M. and A. Triki, On uniformly closed ideals in /-algebras, 2nd Conference, Functions spaces, Marcel Dekker (1995), 29-33. [13] Bernau, S. and Huijsmans, C. B., Almost /-algebras and d-algebras, Math. Proc. Cambridge. Phil. Soc. 107 (1990), 287-308. [14] Bernau, S. J. and C. B. Huijsmans, The order bidual of Almost /-algebras and d-algebras, Trans. Amer. Math. Soc. 347 (1995), 4259-4274. [15] Bernau, S. J. and C. B. Huijsmans, The Schwarz inequality in Archimedean /-algebras, Indag. Math. 7 (1996), 137-148. [16] Beukers, F. and C. B. Huijsmans, Calculus in /-algebras, J. Austral. Math. Soc. (Ser. A) 37 (1984), 110-116 [17] Beukers, F., C. B. Huijsmans and B. de Pagter, Unital embedding and complexification of /-algebras, Math, Z., 183 (1983), 131-144. [18] Bigard, A. and K. Keimel, Surles endomorphismes conservant les polaires d'un groupe reticule archimedien, Bull. Soc. Math. France, 97 (1969), 381-398. [19] Bigard, A., K. Keimel and S. Wolfenstein, Groupes et Anneaux Reticules. Lecture Notes in Math. vol. 608, Springer Verlag, 1977. [20] Birkhoff, G., Lattice ordered groups, Annals of Math., 43 (1942), 298-331 [21] Birkhoff, G., Lattice Theory, 3rd Edition. Amer. Math. Soc. Colloq. Pub!. no. 25, American Mathematical Society 1967. [22] Birkhoff, G. and R. S. Pierce. Lattice-ordered rings, An. Acad. Brasil. Cienc. 28 (1956), 41-69 [23] Boulabiar, K., A relationship between two almost /-algebra products, Algebra Univ., 43 (2000), 347-367 [24] Boulabiar, K., Products in almost /-algebras, Comment. Math. Univ. Carolinae, 41 (2000), 747-759. [25] Boulabiar, K., A representation theorem ford-rings, Submitted. [26] Boulabiar, K., Positive derivations on almost /-rings, To appear in Order. [27] Boulabiar, K., On positive orthosymmetric bilinear maps, Submitted. [28] Boulabiar, K. and E. Chi!, On the structure of almost /-algebras, Demonstratio Math., 34 (2001), 749-760. [29] Boulabiar, K. and M. A. Toumi, Lattice bimorphisms on /-algebras, Algebra Univ. 48 (2002), 103-116. [30] Boulabiar, K., Buskes, G. and Henriksen, M., A Generalization of a Theorem on Biseparating Maps, To appear in J. Math. Ana. Appl.



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[31] Buskes, G., B. de Pagter and A. van Rooij, Functional calculus on Riesz spaces, Indag. Math., 2 (1991), 423-436. [32] Buskes, G. and A. van Rooij, Riesz spaces and Ultrafilter Theorem, Compositio Math. 83 (1992), 311-327. [33] Buskes, G. and A. van Rooij, Small Riesz spaces, Math. Proc. Cambridge Phil. Soc., 105 (1989), 523-536. [34] Buskes, G. and A. van Rooij, Almost /-algebras: commutativity and Cauchy-Schwarz inequality, Positivity, 4 (2000), 227-231. [35] Buskes, G. and A. van Rooij, Almost /-algebras: structure and the Dedekind completion, Positivity, 4 (2000), 233-243. [36] Buskes, G. and A. van Rooij, Squares of Riesz spaces, Rocky Mountain J. Math. 31 (2001), 45-56. [37] Chi!, E., Structure and Dedekind completion of d-algebras, Submitted. [38] Conrad, P., The additive group of an /-ring. Canad. J. Math. 25 (1974), 1157-1168. [39] Conrad, P. and J. Diem, The ring of polar preserving endomorphisms of an abelian latticeordered group, Illinois J. Math., 17 (1971), 222-240. [40] Colville, P., G. Davis and K. Keimel, Positive derivations on /-rings, J. Austral. Math. Soc. 23 (1977) 371-375. [41] Ellis, A. J., Extreme positive operators, Quart. J. Math. Oxford, 15 (1964), 342-344 [42] Feldman, D. and M. Henriksen, /-Rings, subdirect products of totally ordered rings, and the Prime Ideal, Indag. Math., 50 (1998), 121-126. [43] Fremlin, D. H., Tensor products of Archimedean vector lattices, Amer. J. Math. 94 (1972), 778-798. [44] Frink, 0., MR 13,718c, Review of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts [45] Fuchs, L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford-London-New YorkParis, 1963. [46] Grobler, J. J., Commutativity of the Arens product in lattice ordered algebras, Positivity, 3 (1999), 357-364. [47] Gillman, L. and M. Jerison, Rings of Continuous Functions, Springer Verlag, BerlinHeidelberg-New York, 1976. [48] Hager, A. W., and L. C. Robertson, Representing and ringifying a Riesz space, Symposia Math., 21 (1977), 411-431. [49] Henriksen, M., Semiprime ideals of /-rings, Symposia Math., 21 (1977), 401-409. [50] Henriksen, M., A survey of /-rings and some of their generalizations, Ordered algebraic structures, Kluwer Acad. Publ., Dordrecht, (1997), 1-26. [51] Henriksen, M., and J. R. Isbell, Lattice-ordered rings and functions spaces, Pacific J., 12 (1962), 533-565. [52] Henriksen, M., S. Larson and F. A. Smith, When is every order ideal a ring ideal?, Comment. Math. Univ. Carolinae, 32 (1991) 411-416. [53] Henriksen, M. and D. G. Johnson, On the structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. [54] Henriksen, M. and F. A. Smith, Some properties of positive derivations on /-rings, Ordered fields and real algebraic geometry, Contemp. Math., 8, Amer. Math. Soc., (1982), 175-184. [55] Henriksen, M. and F. A. Smith, A look at biseparating maps from an algebraic point of view, Real Algebraic Geometry and Ordered Structures, Contemp. Math., 253, Amer. Math. Soc., (2000), 125-144. [56] Huijsmans, C. B., The order bidual of lattice ordered algebras II, J. Operator Theory 22 (1989), 277-290 [57] Huijsmans, C. B., Lattice ordered algebras and !-algebras: A survey, in: Studies in Economic Theory 2, Positive Operators, Springer Verlag, Berlin, 1991. [58] Huijsmans, C. B. and B. de Pagter, Ideal theory in /-algebras, Trans. Amer. Math. Soc. 269 (1982), 225-245. [59] Huijsmans, C. B. and B. de Pagter, Subalgebras and Riesz subspaces of an /-algebra, Proc. London, Math. Soc., 48 (1984), 161-174. [60] Huijsmans, C. B. and B. de Pagter, The order bidual of lattice ordered algebras, J. Funct. Anal., 59 (1984), 41-64.


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[61] Huijsmans, C. B. and B. de Pagter, Averaging operators and positive contractive projections, J. Math. Ana. Appl., 113 (1986), 163-184. [62] Ionesco Tulcea, A. and C. Ionesco Tulcea, On the lifting property I, J. Math. Anal. Appl., 3 (1961), 537-546. [63] Jarosz, K., Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc., 33 (1990), 139-144. [64] Johnson, D. G., The completion of an archimedean /-ring, J. London Math. Soc, 40 (1965), 493-493. [65] Kudlacek, V., On some types of £-rings. Sb. Vysoke. Uceni Tech. Brno 1-2 (1962), 179-181. [66] Lambek, J. Lectures on Rings and Modules, Blaisdell, 1966. [67] Luxemburg, W. A. J., Some Aspects of the Theory of Riesz Spaces, Univ. Arkansas Lecture Notes Math. 4, Fayetteville, 1979 [68] Luxemburg, W. A. J., A remark on a paper by D. Feldman and M. Henriksen concerning the definition of /-rings, Nederl. Akad. Wetensch. Indag. Math., 50 (1998), 127-130. [69] Luxemburg, W. A. J. and A. C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971. [70] Lozanovsky, G., The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1973), 45-54. [71] Martinez, J. The maximal ring of quotient /-ring. Algebm Univ., 33 (1995), 355-369. [72] Meyer-Nieberg, P., Banach Lattices, Springer Verlag, Berlin Heidelberg New York, 1991. [73] Misra, P.R., On isomorphism theorems for C(X), Acta. Math. Acad. Sci. Hungar., 39 (1982), 179-180. [74] Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep., 11 (1959), 182-188. [75] Nakano, H., Modern Spectral Theory, Tokyo Math. Book Series II, Maruzen, Tokyo, 1950. [76] de Pagter, B., f-algebms and orthomorphisms (Thesis, Leiden, 1981). [77] Phelps, R. R., Extremal operators and homomorphisms, Trans. Amer. Math. Soc., 108 (1963), 265-274. [78] van Putten, B., Disjunctive linear opemtors and partial multiplication in Riesz spaces, (Thesis, Wageningen, 1980). [79] Quinn, J., Intermediate Riesz spaces, Pacific J. Math., 56 (1975), 225-263. [80] Scheffold, E., FF-Banachverbandsalgebren, Math. Z., 177 (1981), 193-205. [81] Scheffold, E., Der Bidual von F-Banachverbandsalgebren, Acta Sci. Math., 55 (1991), 167179. [82] Scheffold, E., Uber Bimorphismen und das Arens-Produkt bei kommutativen DBanachverbandsalgebren, Preprint. [83] Scheffold, E., Uber den ordnungsstetigen Bidual von FF-Banachverbandsalgebren, Arch. Math., 60 (1993), 473-477. [84] Steinberg, S. A., Quotient rings of a class of lattice-ordered rings, Can. J. Math., 25 (1973), 627-645. [85] Steinberg, Stuart A. On the unitability of a class of partially ordered rings that have squares positive. J. Algebra 100 (1986), no. 2, 325-343. [86] Steinberg, Stuart A. On lattice-ordered algebras that satisfy polynomial identities. Ordered algebraic structures (Cincinnati, Ohio, 1982), 179-187, Lecture Notes in Pure and Appl. Math., 99, Dekker, New York, 1985. [87] Steinberg, Stuart A. Unital $!$-prime lattice-ordered rings with polynomial constraints are domains. Trans. Amer. Math. Soc. 276 (1983), no. 1, 145-164. [88] Steinberg, Stuart A. On lattice-ordered rings in which the square of every element is positive. J. Austral. Math. Soc. Ser. A 22 (1976), no. 3, 362-370. [89] Toumi, M. A., On some /-subalgebras of d-algebras, To appear in Math. Reports [90] Triki, A., Stable /-algebras, Algebm Univ., 44 (2000) 65-86. [91] Triki, A., On algebra homomorphisms in complex almost /-algebras, Comment. Math. Univ. Carolinae, 43 (2002), 23-31. [92] Zaanen, A. C., Riesz Spaces II, North Holland, Amsterdam-New York-Oxford, 1983. [93] Zaanen, A. C., Introduction to Opemtor Theory in Riesz Spaces, Springer Verlag, Berlin, 1997. [94] Zelazko, W., Banach algebms, Elsevier, Amsterdam-London-New York, 1973.



133

DEPARTEMENT DU CYCLE AGREGATIF, lNSTITUT PREPARATOIRE AUX ETUDES SCIENTIFIQUES ET TECHNIQUES, UNIVERSITE DU 7 NOVEMBRE

A CARTHAGE,

BP 51, 2070-LA MARSA, TUNISIA

E-mail address: karim. boulabiar.. = ,\. By assumption there is an element {b.>.} in £00 (ReA) such that we have the inequality l{f>..}- {b>.}l < 1/2

on x and hence also on some open subset U of (3(A x Xl) containing calculation we have

xo = {p E (3(A x Xl) ll{f}(p)- f(xo)l

~ ~

x.

By a simple

Vf E CIR(X1)}.

Thus there is a function fa in CIR(Xl) such that the set

{p E (3(A x Xl) ll{fo}(p)- fo(x)l ~ 1/2} is a subset of U. Put Kx

= {y

E X1l lfo(Y)- fo(x)l ~ 1/2},

a compact neighbourhood of x. Then Ax Kx is contained in U so that If>..- b>-1 ~ 1/2 on Kx for all ,\ in A. Let M = sup>. II b>. II, a finite quantity because (b>.) is in £00 (Re A). Take any f in Ca(X, JR) with II f lloo~ 1. By induction we construct a sequence (bn) of elements from ReA, with II bn II~ M for all n, such that n-1

12n(f- LTibi) -bnl < ~ i=O

on Kx. The function b = I::o 2-ibi is in Band b =

f

on Kx. 0

PROPOSITION 3. Let x 0 E X and suppose that there is an open neighbourhood X 0 of x 0 such that every function in C0 (X, JR), which vanishes outside X 0 can be unifromly approximated on X by elements from ReA. Then £00 (ReA) separates the points of xo.

Proof. Suppose p,q are in x0. There is an element {f>..} in £00 (C0 (X,JR)), where each f>.. vanishes outside X 0 , such that {f>..}(p) = 0 and {f>..}(q) = 1. By assumption we can for each ,\ E A find a function b;., in ReA with II f>..- b>. lloo~ Thus we have {b>.}(P) ~ and {b.>.}(q) 2 ~ since II{!>.}- {b>.}lloo,{J(AxXI) ~ Let a>. = b>. + ic>. be in A for each .A. Now, the A-net {ea"' - 1} is in £=(A) and it separates p from q since {ean} does. Since

:!· :!·

:!

l{ea"'}(p)l

= {eb"'}(p)

~ e1/4 and l{ea"'}(q)l

= {eb"'}(q) 2 e3/4,

we see that £=(A) separates p from q and thus £ (ReA) does so as well. 0 00


EGGERT BRIEM

142

We also need the aforementioned extension of the Stone-Weierstrass theorem due to de Leeuw and Katznelson (see [4], Theorem 4.21). We can prove a version of this result, Proposition 4, in the same way as the original one, the proof is omitted. PROPOSITION 4. Let Y be a compact Hausdorff space and B a subspace ofO~.(Y) which separates the points of Y and contains the constant functions. Suppose that B is also a normed space with the norm II· liB which dominates the uniform norm. Let h be a continuous function defined on an interval I. Suppose that h is nonaffine in every neighbourhood of an interior point t 0 in I. Suppose that there exists a positive real number r5 > 0 with (to- r5, t 0 + r5) -}

E cl(g=(Re A)),

the uniform closure of goo(ReA), if {c>-} E goo(ReA) and II {c>-} II< r5. Take xo E E such that b(x 0 ) = t 0 , an interior point of b(E). We restrict to x0 and deduce that h o {to+ c>-} E cl(goo(ReA)Ixo),

for every {c>-} E goo(ReA)Ixo whose quotient norm satisfies II {c>-} II< r5. Since {c} is constant on x0 for any c E ReA, the space goo (ReA) lxo contains the constant functions, it also separates the points of x0 by Proposition 2 and 3. Then Proposition 4 shows that goo(ReA)Ixo is dense in CJR.(x 0 ) and thus, by the local version of Bernard's Lemma, ReAIK = CJR.(K) for some compact neighbourhood K of x 0 . The theorem of Sidney and Stout, [9], then shows that AIK = C(K). By Proposition 2, there exists an open neighbourhood X 0 of x 0 such that every f E C 0 (X, ~) which vanishes outside X 0 can be approximated uniformly on X by


ALGEBRAS ON LOCALLY COMPACT SPACES

I43

elements of ReA. We may assume K C X 0 . Let KI be a compact neighbourhood of x 0 such that KI is in the interior of K. Since E is antisymmetric and contains more than one point, En KI contains more than one point. On the other hand, we will show that {f E Co(X,JR) If= 0 outside KI} C AI. It will follow that E n K I = {x 0 }, which will be a contradiction proving A = Co(X, JR). Let f E Co(X, JR) with f = 0 outside KI. A function u E Co(X, JR) with lui : : ; 1 on X, u = 1 on KI, and u = 0 outside K can be uniformly approximated by functions in ReA. Thus, for every positive integer n, there exists bn E ReA such that 1 - ; 2 ::::; bn ::::; 1 on KI and bn ::::; ; 2 outside K. Without loss of generality we may assume that lbnl : : ; 1 on X. Take Cn E A with Recn = bn and put an= (ecn-I )n. Then an E AI, e--i'> ::::; lanl : : ; 1 on KI and lanl : : ; e-n+-/'; outside K. Since AIIK = C(K) and thus AIIKI = C(KI), there exists a positive real number M such that for every positive integer n there exists gn E AI such that gnan = 1 on KI with llgnlloo : : ; M. Since AIIK = C(K), there is a function a! E AI with a!= f on K. Then afgnan E AI and by a simple calculation llatgnan- flloo-+ 0 as n-+ oo, that is, f E AI. D References [1] A. Bernard, Espaces des parties relies des lments d'une algebre de Banach de fonctions, J. Funct. Anal. 10 (1972), 387-409. [2] E. Briem, Approximations from Subspaces of Co(X), J. Approx. Theory 112 (2001), 279-294. [3] A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc. !969. [4] R.B. Burckel, Characterizations of C(X) among its subalgebras (Lecture Notes in Pure and Appl. Math. 6). Marcel Dekker, New York 1972. [5] 0. Hatori, Functions which operate on the real part of a uniform algebra, Proc. A mer. Math. Soc. 83 (1981), 565-568. [6] 0. Hatori, Separation properties and operating functions on a space of continuous functions, Internat. J. Math. 4 (1993), 551-600. [7] 0. hatori, Range transformations on a Banach function algebra. IV, Proc. A mer. Math. Soc. 116 (1992), 149-156. [8] S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pac. ]. Math. 80 (1979), 265-272. [9] S. J. Sidney and E. L. Stout, A note on interpolation, Proc. Amer. Math. Soc. 19 (1968), 380-382. [10] J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426. SCIENCE INSTITUTE, UNIVERSITY OF ICELAND, REYKJAVIK, ICELAND

E-mail address: briem@hi. is




http://dx.doi.org/10.1090/conm/328/05775 Contemporary Mathematics

Volume 328, 2003

Some mapping properties of p-summing operators with Hilbertian domain Qingying Bu

ABSTRACT. We prove that if H is a Hilbert space, Y is a Banach space and u: H---+ Y is absolutely p-summing for some p :0:: 1, then for any 1 < q < oo, u takes absolutely q-summable sequences in H into members of £q@Y, the projective tensor product of t'q andY.

Given a real or complex Banach space X and c~eak(X) the Banach spaces ll(xn)nllc~tron"(X) = ll(llxnll)nllcv and ll(xn)nllc~eak(X) respectively (cf. [4, pp. 32-36]). For 1 < p < all (strongly p-summable) sequences in X such (x~)n E c:;,eak(X*), normed by g~trong(X)

ll(xn)nllcv(X) =sup

{I~ x~(xn)l

:

and 1 :S p < oo, we denote by of sequences in X with norms = SUPx*EBx• ll(x*xn)nllcv' oo, let £p (X) denote the space of that E~=l lx~(xn)l < oo for each

l (x~)nl c~,eak(X*)

:S

1},

where pt is the conjugate of p, i.e., 1/p + 1/pt = 1. With this norm £p(X) 1s a Banach space (cf. [1, 3]). Note: In [2] it was shown that £p(X) is exactly £p@X, the projective tensor product of £P and X. In this note we use this identification of £p(X) with £p@X to deduce a surprising mapping property of absolutely p-summing operators that have a Hilbert space domain. While the main result of this note can be derived from some by-now famous results of K wapien, it was discovered because of the identification of £P (X) with £p@X, moreover, this identification leads itself to a proof that is a clean and clear application of Khinchin's inequality, Kahane's inequality, and Pietsch's Domination theorem - all fundamental aspects of the theory of p-summing operators. From the definitions, we have for 1 < p < oo, £p(X) ~ g~trong(X) ~ g~eak(X), and I ·llc~eak(X) :S II· llc~tron"(X) :S ll·llcv(X)· Moreover, in case dimX = oo, all the containments are proper. For Banach spaces X and Y and a continuous linear operator u : X ----+ Y, define u: xN ----+ yN by (xn)n f------7 (uxn)n· Then u is a linear operator. Thanks 2000 Mathematics Subject Classification. 46B28.

145

©



QINGYING BU

I46

to the Closed Graph Theorem, each of

u : c;eak (X) ____, c;eak (Y); u : g~tron

9

(X) ____, g~tron

9

u : £P (X) ____, £p (Y)

(Y);

is a continuous operator with

llulle;eak(X)--->f;:'"ak(Y) = llulle~trong(X)--->f~trong(Y)

= llullep(X)--->fp(Y) = llull·

We should mention here Khinchin's inequality (cf. [4, p. 10]) and Kahane's inequality (cf. [4, p. 211]) each of which plays a critical role in this paper. Let rn(t) denote the Rademacher functions (cf. [4, p. 10]), namely, rn: [0, 1]-----+ ffi.,n EN defined by rn(t) := sign(sin2n7Tt).

Khinchin's Inequality. For any 0 < p < oo, there are positive constants Ap, Bp such that for any scalars a I, a 2 , · · · , an, we have

Kahane's Inequality. If 0 < p, q < oo, then there is a constant Kp,q > 0 for which

([II t, c,(L)x,ll' dt)

k/o 0: for all possible c in (1)}.

In 1973, J. S. Cohen [3] introduced strongly p-summing operators between Banach spaces, namely, a Banach space operator u : X -----+ Y is called strongly p-summing operator, 1 < p < oo, if u takes g~trong(X) into £p(Y). Let Dp(X, Y) denote the space of all strongly p-summing operators from a Banach space X to a Banach space Y. If u : X -----+ Y is a strongly p-summing operator then, thanks to the Closed Graph Theorem, u: g~trong(X) -----+ £p(Y) is continuous. So we define a


PROPERTIES OF p-SUMMING OPERATORS

147

strongly p-summing norm Dp(u) on Dp(X, Y) to be Dp(u) = llulle~trong(X)--+lv(Y)· With this norm Dp(X, Y) is a Banach space. There is an equivalent definition of strongly p-summing operators, namely, a Banach space operator u : X ---+ Y is strongly p-summing if and only if there is a constant c > 0 such that for any Xt, x2, · · · , Xn E X, and any yi, Y2, · · · , y~ E Y*,

(2) In this case, Dp(u) = inf{c > 0: for all possible c in (2)}.

Note: Actually, u E Dp(X, Y) means that u takes absolutely p-summable sequences in X into members of lp®Y. Main Theorem. Let 1 < p, q < oo; and let H be a Hilbert space and Y be a Banach space. Then ITp(H, Y) ~ Dq(H, Y), i.e., if u : H ---+ Y is absolutely p-summing, then u takes absolutely q-summable sequences in H into members of fq®Y.

PROOF. First consider H = f2 for n E N. Let u E 1Ip(f2, Y). By Pietsch's Domination Theorem [4, p. 44], there is a regular probability measure f.l on Be 2 such that for any x E f2,

lluxll Now for x1,x2, · · · ,xm E

~

1rp(u) ·

f2,

(

le-q l(x, z)IP dJl(z)

1/p )

(3)

and yi,y2, · · · ,y;,_ E Y*, we have n

Xk =

L Xk,iei,

k = 1,2,··· ,m.

i=1

Then

L I(L Xk,iuei, yj.) I

m

mn

L l(uxk, Yi:)l

k=1

k=1

<
/r

r}

B, (t, l(e;, z)l'

(L,, lizII'

dM(z)

dM(z)) >/r

YIP ..)z for some y E F 1 , z E F2 and>.. E [0, 1]. If >.. is unique, for each x E K \ (F1 U F 2) but independent of y and z, then F 1 and F 2 are said to be parallel faces; parallel faces are automatically norm closed. If in addition y and z are unique then F 1 and F 2 are called split faces. See, for example, [1, 2] for the general theory of compact convex sets and related topics. Let K be a compact convex set. Recall that the facial topology on 8K is given by defining {F

n 8K: F is a closed split face of K}

to be the family of all closed sets. The facial topology is weaker than the relative topology on 8K. The centre Z(A(K)) of A(K) is the set of all those functions in A(K) whose restriction to 8K is facially continuous. The central functions h E Z(A(K)) are characterised by the following property (see, for example, [1, Corollary II.7.4] or [2, Theorem 3.1.4]): for all f E A(K), there exists g E A(K) such that g(x) = h(x)f(x) for all x in 8K. The uniqueness of the continuous affine function g is clear, since a continuous affine function on a compact convex set is completely determined by its values on the extreme boundary, and consequently we may write g = h ·f. In this way it is useful to think of the central functions as the multipliers of A(K). A compact convex set K is a Choquet simplex whenever for all bounded (real) linear functionals 0 the set K n (
0 the set K n (r.p+o:K) is either empty or a singleton, or contains some 'I/J+f3K where 'ljJ E A(K)* and (3 > 0. In particular, every Choquet simplex has the unique decomposition property. A result of Grothendieck [10], see also [5, p. 272], shows that the state space of a (unital) C*-algebra has the unique decomposition property. Thus the results of this paper apply, giving an affine homeomorphism between the state spaces K and S of two C*-algebras whenever the associated (real) affine function spaces A(K) and A(S) are linearly isometric. We give some examples below, the second and third of which show that the unique decomposition property and the condition of Ellis and So are independent geometric properties of a compact convex set. EXAMPLE 2.1. Let K be the state space of the C*-algebra M2 , of all 2 x 2 matrices over C. Then, by Grothendieck's result, K has the unique decomposition property. Also, K is affi.nely homeomorphic to a closed ball in JR3 (see [2, p. 241]) and satisfies the condition of Ellis and So since it has no proper complementary faces. EXAMPLE 2.2. Let K be a triangular hi-simplex in 3-dimensional space as in the figure below, Figure 1. Then no proper face is (geometrically) parallel to any other and hence K n (r.p + o:K) is either empty, a singleton or has non-empty interior, for r.p E A(K)* and o: > 0. It follows, by the geometric characterisation of Ellis, that K has the unique decomposition property. However (F, F') is a pair of complementary faces of K, but not split, and hence K does not satisfy the condition of Ellis and So. EXAMPLE 2.3. Let K be an icosahedron in 3-dimensional space. Then K satisfies the condition of Ellis and So because it has no proper complementary faces. However it does not have the unique decomposition property because it has

F

FIGURE 1. The hi-simplex of Example 2.2


154

AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

faces (geometrically) parallel to each other and hence K interior for some rp E A(K)* and a > 0.

n (rp + aK) can have empty

3. Results Throughout this section K and S will denote compact convex sets of locally convex (Hausdorff) spaces. We will also let T: A(K) ~ A(S) denote a surjective linear isometry. Notice that if T1 = 1 then for rp E A(S)* with II'PII = 1 = rp(1), we have IIT*rpll = II'P 0 Til= II'PII = 1 = rp(1) = rp(T1) = T*rp(1); here T*: A(S)* ~ A(K)* denotes the dual map ofT. Consequently T*(S) = K and hence T* induces an affine homeomorphism 0': S ~ K such that Tf(s) = T*s(f) = f(O'(s)) for all s E S; that is, Tis a composition operator f f--7 f o 0'. In Proposition 3.3 we see that T is a weighted composition operator if and only if T1 is central. To do this we 'decompose' S by defining (3.1)

S1 = {s E S: (T1)(s) = 1}

It is clear that

sl

and

S2 = {s E S: (T1)(s) = -1}.

and s2 are closed faces of s.

LEMMA 3.1. Let S1 and S2 be as in (3.1). Then aSs;;; S1 U S2, and S1 and S2 are closed parallel faces of S. PROOF. Observe that the dual map T* is a linear isometry from A(S)* onto A(K)*. Hence T* maps the extreme points of the closed unit ball of A(S)* onto the extreme points of the closed unit ball of A(K)*. Thus

T*(aS U a( -S)) = aK U a( -K). Consequently, for each s E as we have T* s is in aK or a(- K) and hence

T1(s) = T*s(1) = ±1. Therefore ass;;; S 1 U S 2 and thus by the Krein-Milman TheoremS= co (S1 U S 2), since S, S 1 and S2 are all (weak*) compact. Thus S1 and S 2 are complementary faces since they are clearly disjoint. For each sinS with s =.Ax+ (1- .A)y, where x E S 1, y E S 2, and>. E (0, 1), we have T*s = >.T*x + (1- >.)T*y. Thus,

T1(s) = T*s(1) = >.T*x(1) + (1- >.)T*y(1) = >.- (1- >.) = 2>.- 1. This establishes the uniqueness of>.= ((T1)(s) + 1)/2 and the result follows.

D

We now specialise to the case when K and S have the unique decomposition property. LEMMA 3.2. Let sl and s2 be as in (3.1). Suppose that K has the unique decomposition property. Then S 1 and S 2 are complementary split faces of S. PROOF. By Lemma 3.1, for each s E S\(S 1 U S 2) we may write s = >.x + (1- >.)y where X E sl, y E s2 and 0 < A < 1, and A is unique. We consider the decomposition T* s = >.T*x- (>. -1)T*y. Since X E sl we have T*x E K and hence >.T*x is positive. Similarly, T*y E -K and hence (>.- 1)T*y is positive. Also IIT*sll = 1 = >. + (1- >.) = 11>-T*xll +II(>. -1)T*yll· Thus, by the unique decomposition property, >.T*x and(>.- 1)T*y are unique. By the uniqueness of>., we have T*x and T*y are unique. Since Tis surjective, T* is injective and the result follows. D


THE UNIQUE DECOMPOSITION PROPERTY AND BANACH-STONE THEOREM

155

r- 1 in (3.1) we may 'decompose' K by defining (3.2) K 1 = {k E K: r- 11(k) = 1} and K2 = {k E K: r- 11(k) = -1}. Applying Lemmas 3.1 and 3.2 to r- 1 we see that K 1 and K 2 are complementary By replacing T by

split faces of K whenever 8 has the unique decomposition property. We say that T is a weighted composition operator whenever there exists a central function h in A(8) and a continuous affine mapping a: 8 --+ K such that Tf = h · f o a for all f E A(K); that is, Tf(s) = h(s)f(a(s)) for all s E 88. The following proposition asserts that the linear isometry T is a weighted composition operator, with T1(s) = ±1 on 88, if and only if T1 is central.

PROPOSITION 3.3. LetT: A(K) --+ A(8) be a linear mapping. Then the following are equivalent: a) T is an isometry and T1 is central; b) T is a weighted composition operator of the form T f = h · f o a for all f E A(K) where a is an affine homeomorphism and h(s) = ±1 for all s E 88. PROOF. See [4, Theorem 3.3] or [13].

0

We now apply the above decompositions of K and 8 to prove our main theorem. THEOREM 3.4. Suppose that K and 8 have the unique decomposition property. Then the real affine function spaces A(K) and A(8) are linearly isometric if and only if K and 8 are affinely homeomorphic. Moreover, every linear isometry from A(K) onto A(8) may be written as a weighted composition operator. PROOF. It suffices to show necessity. Suppose that Tis a linear isometry from A(K) onto A(8). We 'decompose' 8 into the complementary split faces 81 and 82 of (3.1) and, similarly, K into the complementary split faces K1 and K2 of (3.2). Since (T- 1)* = (T*)- 1, we have T*(81) = K 1 and T*(82) = -K2, and hence we may define a: 8 --+ K by a(Ax + (1- A)y) = AT*(x)- (1- A)T*(y)

whenever X E 81' y E 82 and 0 ~ A ~ 1. We see that a is an affine homeomorphism from 8 =co (81 U 82) onto K =co (K1 U K2). Moreover, to show that T is a weighted composition operator it suffices, by Proposition 3.3, to show that h = T1 is central. Let f E A(8) then, since a: 8 --+ K is an affine homeomorphism, we may write f =goa for some g E A(K). Note that for X E 81 we have Tg(x) = T*x(g) = g(a(x)) = h(x)f(x). Similarly for X E 82 we have Tg(x) = T*x(g) = -g(a(x)) = h(x)f(x). Therefore, for all x E 88 ~ 81 U 82 we have Tg(x) = h(x)f(x), and the result follows. 0 References [1] E.M. Alfsen, Compact convex sets and boundary Integmls, Ergebnisse der Mathematik, 57, (Springer-Verlag, Berlin-Heidelberg-New York, 1971). [2] L. Asimow and A.J. Ellis, Convexity theory and its applications in functional analysis, London Math. Soc. Monograph, 16 (Academic Press, London, 1980). [3] E. Behrends, M -Structure and the Banach-Stone Theorem, Lecture Notes in Mathematics 736, (Springer-Verlag, Berlin-Heidelberg-New York, 1979). [4] A. Curnock, J. Howroyd, and N.-C. Wong, Isometries of affine function spaces, preprint. [5] J. Dixmier, C*-Algebms (North-Holland Publishing Co., Amsterdem-New York-Oxford, 1982).


156

AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

[6] A.J. Ellis, An intersection property for state spaces, J. London Math. Soc., 43 (1968), 173176. [7] A.J. Ellis, Minimal decompositions in partially ordered normed spaces, Proc. Camb. Phil. Soc., 64 (1968), 989-1000. [8] A.J. Ellis, On partial orderings of normed spaces, Math. Scand., 23 (1968), 123-132. [9] A.J. Ellis and W.S. So, Isometries and the complex state spaces of uniform algebras, Math. z., 195 (1987), 119-125. [10] A. Grothendieck, Un resultat sur le dual d'une C*-algebre, J. Math. Pures Appl., 36 (1957), 97-108. [11] A.J. Lazar, Affine products of simplexes, Math. Scand., 22 (1968), 165-175. [12] R.R. Phelps, Lectures on Choquet's Theorem, Second Edition, Lecture notes in Mathematics 1757 (Springer-Verlag, Berlin, 2001). [13] T.S.R.K. Rao, Isometries of Ac(K), Proc. Amer. Math. Soc., 85 (1982), 544-546. SCHOOL OF COMPUTING, INFORMATION SYSTEMS AND MATHEMATICS, SOUTH BANK UNIVERSITY, LONDON SE1 OAA, ENGLAND. E-mail address: curnoca@sbu. ac. uk DEPARTMENT OF MATHEMATICAL SCIENCES, GOLDSMITHS COLLEGE, UNIVERSITY OF LONDON, LONDON SE14 6NW, ENGLAND. E-mail address: [email protected] DEPARTMENT OF APPLIED MATHEMATICS, NATIONAL SUN YAT-SEN UNIVERSITY, KAOHSIUNG 80424, TAIWAN, R.O.C. E-mail address: wong@math. nsysu. edu. tw



A Survey of Algebraic Extensions of Commutative, Unital Normed Algebras Thomas Dawson ABSTRACT. We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.

Introduction Algebraic extensions have had striking applications in the theory of uniform algebras ever since Cole used them (in [5]) to construct a counterexample to the peak-point conjecture. Apart from this, their main use has been in (a) the construction of examples of general, normed algebras with special properties and (b) the Galois theory of Banach algebras. We shall not discuss (b) here; a summary of some of this work is included in [29]. In the first section of this article we shall introduce the types of extensions and relate their applications. The section ends by giving the exact relationship between the types of extensions. Section 2 contains a table summarising what is known about the extensions' properties. A theme lying behind all the work to be discussed is the following question:

(Q) Suppose the normed algebra B is related to a subalgebra A by some specific property or construction. (For example, B might be integral over A: every element bE B satisfies ao+ · · ·+an-1bn- 1+bn = 0 for some ao, ... , an-1 EA.) What properties of A (for example, completeness or semisimplicity) must be shared by B? This is a natural question, and interesting in its own right. Many special cases of it have been studied in the literature. We shall review the related body of work in which B is constructed from A by adjoining roots of monic polynomial equations. Throughout this article, A denotes a commutative, unital normed algebra, and A its completion. The fundamental construction of [1] applies to this class of 1991 Mathematics Subject Classification. Primary 46J05, 46J10. This research was supported by the EPSRC. © 2003 American Mathematical Society 157


THOMAS DAWSON

158

algebras. Algebraic extensions of more general types of topological algebras have received limited attention in the literature (see [19], [21]). If E is a subset of a ring then (E) will stand for the the ideal generated by E. 1. Types of Algebraic Extensions and their Applications 1.1. Arens-Roffman Extensions. Let o:(x) = a 0 + · · · + an_ 1xn- 1 + xn be a monic polynomial over the algebra A. The basic construction arising from A and o:(x) is the Arens-Roffman extension, Aa· This was introduced in [1]. Most of the obvious questions of the type (Q) for Arens-Roffman extensions were dealt with in this paper and in the subsequent work of Lindberg ([18], [20], [13]). See columns two and three of Table 2.2. All the constructions we shall meet are built out of Arens-Roffman extensions. DEFINITION 1.1.1. A mapping B: A -+ B between algebras A and B is called unital if it sends the identity of A to the identity of B. An extension of A is a commutative, unital normed algebra, B, together with a unital, isometric monomorphism B:A-+ B.

The Arens-Roffman extension of A with respect to o:(x) is the algebra A"' := A[x]/(o:(x)) under a certain norm; the embedding is given by the map v: a f-+ (o:(x)) +a. To simplify notation, we shall let x denote the coset of x and often omit the indeterminate when using a polynomial as an index. It is a purely algebraic fact that each element of A"' has a unique representative of degree less than n, the degree of o:(x). Arens and Hoffman proved that, provided the positive number t satisfies the inequality tn ;::: 2.::::~:~ llak II tk, then (bo, ... , bn-1 E A)

defines an algebra norm on A"'. The first proposition shows that Arens-Roffman extensions satisfy a certain universal property which is very useful when investigating algebraic extensions. It is not specially stated anywhere in the literature; it seems to be taken as obvious. 1.1.2. Let A( 1) be a normed algebra and let B: A( 1)-+ B( 2) be a unital homomorphism of normed algebras. Let o: 1 ( x) = ao + · · · + an-1 xn- 1 + xn E A(ll[x] and B( 1 ) = A~ 1 (. Let y E B( 2 ) be a root of the polynomial o: 2 (x) := B(o:I)(x) := B(a0 ) + · · ·+ B(an_ 1)xn- 1 + xn. Then there is a unique homomorphism ¢>: B( 1 ) -+ B( 2 ) such that PROPOSITION

r

B(1) v

~

B(2)

/&

is commutative and ¢>(x)

= y.

A(1)

The map ¢> is continuous if and only if() is continuous. PROOF.

This is elementary; see

[7]


0

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

159

1.2. Incomplete Normed Algebras. A minor source of applications of Arens-Roffman extensions fits in nicely with our thematic question (Q): these extensions are useful in constructing examples to show that taking the completion of A need not preserve certain properties of A. The method uses the fact that the actions of forming completions and ArensRoffman extensions commute in a natural sense. A special case of this is stated in [17]; the general case is proved in [7], Theorem 3.13, and follows easily from Proposition 1.1.2. It is convenient to introduce some more notation and terminology here. Let O(A) denote the space of continuous epimorphisms A~ C; when 0 appears on its own it will refer to A. As discussed in [1], this space, with the weak *-topology relative to the topological dual of A, generalises the notion of the maximal ideal space of a Banach algebra. In fact, it is easy to check that 0 is homeomorphic to O(A), the maximal ideal space of the completion of A. The Gelfand transform of an element a E A is defined by

a: 0

~

C; w f---+ w(a)

and the map sending a to a is a homomorphism, r, of A into the algebra, C(O), of all continuous, complex-valued functions on the compact, Hausdorff space 0. We denote the image of r by A. A good reference for Gelfand theory is Chapter three of [24]. DEFINITION

is injective.

1.2.1 ([1]). The algebra A is called topologically semisimple if

r

If A is a Banach algebra then this condition is equivalent to the usual notion of semisimplicity. The precise conditions under which Aa is topologically semisimple if A is are determined in [1]. In [17] Lindberg shows that the completion of a topologically semisimple algebra need not be semisimple. In order to illustrate Lindberg's strategy we recall two standard properties of normed algebras. DEFINITION 1.2.2. The normed algebra A is called regular if for each closed subset E ~ 0 and wE 0-E there exists a E A such that a(E) ~ {0} and a(w) = 1. The algebra is called local if A contains every complex function, f, on 0 such that every wE 0 has a neighbourhood, V, and an element a E A such that flv = arv.

It is a standard fact that regularity is stronger than localness; see Lemma 7.2.8 of [24]. EXAMPLE 1.2.3. Let A be the algebra of all continuous, piecewise polynomial functions on the unit interval, I, and a(x) = x 2 - idr E A[x]. Let A have the supremum norm. By the Stone-Weierstrass theorem, A= C(I) and hence 0 is identifiable with I. Clearly A is regular. We leave it as an exercise for the reader to find examples to show that Aa is not local. This is not hard; it may be helpful to know that in this example the space O(Aa) is homeomorphic to { (s, >.) E I x C: >. 2 = s }. This follows from facts in [1]. In the present example, neither localness nor regularity is preserved by (incomplete) Arens-Roffman extensions.


THOMAS DAWSON

160

Finally we can explain the method for showing that some properties of normed algebras are not shared by their completions because, in the above, 'non-regularity' is not preserved by completion of Aa (nor is 'non-localness'). To see this, note that A is clearly regular if A is and so by a theorem of Lindberg (see Table 2.2) the ArensRoffman extension (A)a is regular. But, by a result of [17] referred to above, this algebra is isometrically isomorphic to the completion of Aa. Of course Lindberg's original application was much more significant; there are simpler examples of the present result: for example the algebra of polynomials on I.

1.3. Uniform Algebras. It is curious that the application of Arens-Roffman extensions to the construction of integrally closed extensions of normed algebras did not appear in the literature for some time after [1]. It was seventeen years later until a construction was given in [22]. Even then the author acknowledges that the constuction was prompted by the work of Cole, [5], in the theory of uniform algebras. Cole invented a method of adjoining square roots of elements to uniform algebras. He used it to extend uniform algebras to ones which contain square roots for all of their elements. Apart from feeding back into the general theory of commutative Banach algebras (mainly accomplished in [22] and [23]) his construction provided important examples in the theory of uniform algebras. We shall describe these after recalling some basic definitions. DEFINITION 1.3.1. A uniform algebra, A, is a subalgebra of C(X) for some compact, Hausdorff space X such that A is closed with respect to the supremum norm, separates the points of X, and contains the constant functions. We speak of 'the uniform algebra (A, X)'. The uniform algebra is natural if all of its homomorphisms w E 0 are given by evaluation at points of X, and it is called trivial if

A= C(X).

Introductions to uniform algebras can be found in [4], [11], [26], and [16]. An important question in this area is which properties of (A, X) force A to be trivial. For example it is sufficient that A be self-adjoint, by the Stone-Weierstrass theorem. In [5] an example is given of a non-trivial uniform algebra, (B, X), which is natural and such that every point of X is a 'peak-point'. It had previously been conjectured that no such algebra existed. We shall describe the use of Cole's construction in the next section, but now we reveal some of the detail. PROPOSITION 1.3.2 ([5],[7]). Let U be a set of monic polynomials over the uniform algebra (A, X). There exists a uniform algebra (Au, xu) and a continuous, open surjection 1r: xu -+ X such that (i) the adjoint map 1r*: C(X) -+ C(Xu) induces an isometric, unital monomorphism A-+ Au, and (ii) for every a: E U the polynomial1r*(a:)(x) E Au[x] has a root Pa E Au. PROOF. We let xu be the subset of X x !fY consisting of the elements (~~:,.A.) such that for all a: E U

;(a)(~~:)+

J0

... + J(a) (~~:).A.n(a)-1 n(a)-1 a

+ ,\n(a) a


=0

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

where a(x)

= ;(a)+···+ f(a)

n(a)-1

JO

xn(a)- 1 + xn(a)

161

E U. The reader can easily check

that xu is a compact, Hausdorff space in the relative product topology and so the following functions are continuous:

1r: Xu --+ X; ( K:, .A) ~--'

K:

(a E U). The extension Au is defined to be the closed subalgebra of C(Xu) generated by

1r*(A) U {Pa: a E U} where 1r* is the adjoint map C(X)--+ C(Xu); g ~--'go

1r.

It is not hard to check that Au is a uniform algebra on xu with the required

properties.

D

We shall call Au the Cole extension of A by U. Cole gave the construction for the case in which every element of U is of the form x 2 - f for some f E A. It is remarked in [22] that similar methods can be used for the general case; these were indf)pendently, explicitly given in [7]. By repeating this construction, using transfinite induction, one can generate uniform algebras which are integrally closed extensions of A. Full details of this, including references and the required facts on ordinal numbers and direct limits of normed algebras, can be found in [7]. Again this closely follows [5]. Informally the construction is as follows. Let v be a non-zero ordinal number. Set (A 0 , Xo) = (A, X). For ordinal numbers T with 0 < T :::; v we define

(An XT)

=

{

' xl;)a)

(A~"" l~

(

if T

=

a

+1

and

* ) ) (A,.,X,. ) a llx- Yll whenever x =f. y. We will show that even though some of these sets do not support a nonexpansive right shift, they do support nonexpansive fixed point free mappings. To simplify our computations we will only consider the sets K'Y where the sequence 'Y = bn)n is in (0, 1), is strictly increasing and satisfies 1- 'Yn < 4-n for all n EN. EXAMPLE 2.1. Let I be the identity mapping on K'Y, let T be the right shift defined above, T 2 =ToT, T 3 =ToTo T, and so on. DefineR: K'Y--+ K'Y by

R :=~I+

212 T + 2\ T2 + 214 T3 +... .

A simple calculation shows that if x = b1t1, 'Y2t2, "(3t3, ... ) E K'Y, then

R(x) = (11(~t1

+ ~),12(~t2

+ ~t1 + ~),"(3(~t3

+ ~t2 + ~t1 + ~), ... ).

It is easily seen that if R(x) = x, then tn = 1 for all n E N, and thus x is not an element of K'Y; that is, R is fixed point free on K'Y. To see that R is nonexpansive on K'Y, let x = bntn)n andy= ("fnSn)n be elements of K'Y" Then, 1 + -4 1 + · · · + 2n1 ) < 1 for each n EN. since 1- 4-n < 'Yn < 1, we have 'Yn( -2'Yn 'Yn-1 'Yl Consequently, IIR(x)- R(y)ll

sup {'Ynl~(tn-

Sn) + ~(tn-1-

Bn-1) + · · · + 2 ~ (t1- sl)l}

< sup { 'Yn(~ltn-

snl + ~ltn-1-

Bn-11 + · · · + 2 ~ lh- s11)}

n n

1- + - 1- + · · · + -J-) max 'Yilti- sil} < sup {'Yn(n 2"fn 4'Yn-1 2 'Yl 1~i~n < SUP'Ynltn- snl n

llx-yll· Thus R is a nonexpansive mapping on K'Y.


I73

EXAMPLES OF FIXED POINT FREE MAPS

The mapping R, given in example 201, is an affine mapping on K'Yo Our next example is non-affine on K'Yo EXAMPLE 2020 For an element x = (xn)n in K'Y, we denote by ('yi, x) the sequence ('yi, XI, x2, x3, 00 0)o We define a mappingS : K'Y ---. c0 by S(x) := x, for each x E K'Y, where x = (xi, x 2, x 2, ooo) is the decreasing rearrangement of the sequence ('yi, x); that is, x = ('yi, x) * Note that 0

Also XI 2: X2 2: X3 2: 0 0 0 2: 0 and 0 < 'YI < 1'2 < '/'3 < 0 00 < 1. Therefore fu. > ~ > .fu > ooo > Oo Since fu. = max('Y1 ,x) > 1 S(x) does not necessarily belong n-~-~n n -' to KT However, the mapping S is nonexpansive on K'Y because the operation of decreasing rearrangement is nonexpansive on c0 , so for all x andy in K'Y, we have IIS(x)- S(y)ll

= llx- 1111 = II('YI,x)*- ('YI, y)*ll

: XI 2: '/'I, there exists a unique k E N such that Xk+I < 'Yk+I and Xk 2: '/'ko Thus, for all mEN with m > k, we have, k +XI

+ 000 + Xm+I >

>

(xi + 000+ xk) +(xi (xi+ 000+ xk) +

+ 000+ Xm+d

('YI 1\ XI + 00 0 + 'Yk 1\ Xk + 'Yk+l 1\ Xk+I + 0 00 + 'Ym+l 1\ Xm+I) (x 1 + 0 o0+ xk) + ('yi + ooo+ 'Yk + xk+I + 0oo+ xm+I) ('yi + ooo+ 'Yk) +(xi+ 000 + xk + xk+I + 000+ xm+I) ('yi + 000 + 'Yk) +(xi+ 000 + Xm+d ('/'I + ooo+ '/'k) + ('/'I +XI + ooo+ Xm)


0

P.N. DOWLING,

174

C.J.

It follows that for all mEN with m

LENNARD, AND

TURETT

B.

> k, k

xm+1

>

(1'1

>

'Y1 -

+ · · · + 'Yk)- k + -r1 = -r1- L:U- -r1 )

L: (1 - -r1 ) ;::::

L: 4-

00

00

'Y1 -

j=1

j=1

.

J

= -r1 -

j=1

1

1

3 > 3.

This contradicts the fact that x E c0 and so completes the proof that U is fixed point free on K'Y.

3. The fixed point property in £ 1 [0, 1] One of the most notable works in metric fixed point theory is the construction of Alspach [1] of a non-empty weakly compact convex subset of £ 1 [0, 1] which fails the fixed point property. We begin this section by recalling some of the details of Alspach's construction. Let C := {! E £ 1 [0, 1] : 0 ~ f(t) ~ 1, for all t E [0, 1]}. Now define T: C--+ C by

T f(t) := {min{2f(2t), 1} max{2f(2t- 1)- 1, 0}

for 0 ~ t ~ ~ for ~ < t ~ 1.

for all f E C. Alspach showed that the mapping T is an isometry on C which has two fixed points; namely 0 and X[o, 1]· Alspach also showed that T is an isometric self map of the closed convex subset Co := {! E C: J01 f dm = 1/2} of C, such that Tis fixed point free on C0 . Here, m denotes Lebesgue measure. We now follow a modification of Alspach's example due to Sine [7]. Define S : C --+ C by S(f) := X[o, 1] - f, for all f E C. The mapping S is clearly an isometry of C onto C. Thus the mapping ST is a nonexpansive mapping on C. Sine proved that ST is fixed point free on C. In [5], Llorens-Fuster and Sims prove that a closed bounded convex subset of c0 with non-empty interior fails the fixed point property. We will use the above construction of Alspach, and modification by Sine, to prove a result analogous to the Llorens-Fuster and Sims result in the setting of £ 1 [0, 1]. Specifically we prove the following result. THEOREM 3.1. Let K be a closed, bounded, convex subset of £ 1 [0, 1] with nonempty interior. Then K fails the fixed point property for nonexpansive mappings. PROOF. By translating and scaling, we can assume that K contains the unit ball of £ 1 [0, 1]. Consequently, the set C, constructed above, is a subset of K. Define the mapping R : K --+ K by

Rf(t) := min{lf(t)l, 1}, for 0

~

t

~

1, for all f E K.

It is easily seen that R is a nonexpansive mapping on K and R(f) E C for all --+ K by

f E K. Now define U : K

U(f) := ST(R(f)), for all

f E K.

The mapping U is nonexpansive since all of the mappings, R, S, and T are nonexpansive.


EXAMPLES OF FIXED POINT FREE MAPS

175

We now show that U is fixed point free. Suppose that f E K is a fixed point of U, that is, U(f) = f. Since f E K, R(f) E C, and since ST maps C into C, f = U(f) = ST(R(f)) E C. Note that the mapping R restricted to C is the identity on C. Therefore, f = ST(R(f)) = ST(f) and so f is a fixed point of ST in C. This contradicts Sine's result that ST has no fixed point in C [7], and thus the proof is complete. 0 THEOREM 3.2. Let K be a closed, bounded, convex subset of L 1[0, 1] that contains an order interval [h,g] := {f E L 1 [0, 1] : h ::::; f ::::; g a.e.}, for some h, g E L 1 [0, 1] with h ::::; g and h -::/:- g. Then K fails the fixed point property for nonexpansive mappings. PROOF. By translating by -h, we may assume that h = 0 and g 2: 0 a.e. with g non-trivial. Next, note that there exists a real number c > 0 and a measurable set E with Lebesgue measure m(E) > 0, such that g 2: CXE· By rescaling K by 1/c, we may assume without loss that c = 1. Now, define the mapping R: K __, [0, XE] ~ K by R(f) := If I/\ XE· Note that R is nonexpansive and R equals the identity on [0, XE]· At this stage, consider E. There exists t 0 in the interval [0, 1] such that m( En [0, t 0 ]) = ~ m(E). Let E1,1 := En [0, to] and E1,2 := En (to, 1]. Clearly E is the disjoint union of E1,1 and E1,2 and m(E1,1) = m(E1,2) = ~ m(E). Proceed iteratively from here. Similarly to above, there exist pairwise disjoint measurable subsets E2,1, E2,2, E2,3 and E2,4 of [0, 1] such that E1,1 = E2,1 U E2,2, E1,2 = E2,3 U E 2,4, and each m(E2,k) = m(E). Repeating this construction inductively, we produce a family of measurable subsets (E0 , 1 := E, En,k : n E N, k E {1, ... , 2n}) of [0, 1] such that (XEn,k)n,k is a dyadic tree in L 1[0, 1]. Moreover, letting the measure v be defined on the measurable subsets of E by v = (1/m(E))m, it follows that the Banach space L 1( E, v) is isometrically isomorphic to L 1( [0, 1], m) = L 1[0, 1] via the mapping Z defined as follows:

i

for each XEn,k. Then Z is extended to L := the linear span of the functions XEn,k in the usual way. Of course, Z is an isometry on L. Finally, since L is dense in L 1 (E, v), with dense range in L 1[0, 1], Z extends to a linear isometry from L 1 (E, v) onto L 1[0, 1]. Let W := ST be Sine's variation on Alspach's example, as described above, and note that W maps the order interval C := [0, X[o, 1J] into C. Let's use W to define ~ : [0, XE] --; [0, XEL by~ := z- 1 w z. We have that ~ is a fixed point free L 1[0, 1]-isometry on [0, XE]· Finally, we define U : K __, [0, XE] ~ K via U := ~ R. In a manner analogous to the argument in the proof of Theorem 3.1 above, we see that U is a fixed point free L 1[0, 1]-nonexpansive mapping on K. 0 REMARK 3.3. In [6], Maurey proved that closed, bounded, convex, non-empty subsets of reflexive subspaces of L 1[0, 1] have the fixed point property for nonexpansive mappings. Consequently, Maurey's result, in tandem with Theorem 3.2, shows that reflexive subspaces of L 1[0, 1] cannot contain a non-trivial order interval. In fact, as pointed out by the referee, the argument in the proof of Theorem 3.2 shows


176

P.N. DOWLING, C.J. LENNARD, AND B. TURETT

that infinite-dimensional subspaces of L 1 [0, 1] which contain non-trivial order intervals actually contain isometric copies of L 1 [0, 1] and thus are nonreflexive. The authors thank the referee for his/her comments. References 1. D. Alspach, A fixed point free nonexpansive mapping, Proc. Amer. Math. Soc., 82 (1981), 423-424. 2. P.N. Dowling, C.J. Lennard and B. Turett, Characterizations of weakly compact sets and new fixed point free maps in co, to appear in Studia Math. 3. P.N. Dowling, C.J. Lennard and B. Turett, Weak compactness is equivalent to the fixed point property in co, preprint 4. Kazimierz Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990 5. Enrique Llorens-Fuster and Brailey Sims, The fixed point property in co, Canad. Math. Bull. 41 (1998), no. 2, 413-422. 6. B. Maurey, Points fixes des contractions de certains faiblement compacts de L 1 , Seminaire d'Analyse Fonctionelle, 1980-1981, Centre de Mathematiques, Ecole Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19. 7. R. Sine, Remarks on an example of Alspach, Nonlinear Anal. and Appl., Marcel Dekker, (1981), 237-241. DEPARTMENT OF MATHEMATICS AND STATISTICS, MIAMI UNIVERSITY, OXFORD, OH

E-mail address: dowlinpn@muohio. edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA

E-mail address: lennard+@pi tt. edu

45056

15260

DEPARTMENT OF MATHEMATICS AND STATISTICS, OAKLAND UNIVERSITY, ROCHESTER, MI

48309 E-mail address: turett@oakland. edu



Homotopic composition operators on H 00 (En) Pamela Gorkin, Raymond Mortini, and Daniel Suarez ABSTRACT. We characterize the path components of composition operators on

H 00 (Bn), where Bn is the unit ball of en. We give a geometrical equivalence for the compactness of the difference of two of such operators. For n = 1, we give a characterization of the path components of the algebra endomorphisms.

1. Introduction

Consider the Hardy space H 2 on the unit disk D. Littlewood's subordination principle tells us that for an analytic self-map ¢ of D and a function f in H 2 , the function f o ¢ is once again in H 2 . Thus one defines the composition operator C on H 2 by C¢(!) = f o ¢. The interplay of operator theory and function theory leads to several interesting results. One of these results is Berkson's theorem on isolation of composition operators (see [1] and [14]): THEOREM 1 (Berkson). Let ¢ be an analytic self-map of D. If¢ has radial limits of modulus one on a set E of positive measure, then for every other analytic self-map 'ljJ of D, the following estimate holds:

IIC- C.pll

~

Jmea;(E),

where C and C.p are the corresponding composition operators on H 2 •

Thus, Berkson's theorem tells us that every such operator is isolated in the set of composition operators in the operator norm topology. For example, the identity operator, Cz, is at least a distance of from every other composition operator on H 2 (as is C, where ¢is any inner function). However, not every composition operator is isolated. If ¢ is analytic and ¢ : D ---> sD for some s with 0 < s < 1, then it is easy to check that

Jl72

lim

r->1

IICr- Cll

=

0.

Thus, C¢ is not isolated. Inner functions induce highly noncompact operators, as well as isolated operators. The operators C for which ¢(D) is contained in sD for somes with 0 < s < 1 2000 Mathematics Subject Classification. Primary 47B33; Secondary 47B38. Key words and phrases. composition operator, path components, compact differences.

177

@ 2003 American Mathematical Society


178

PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

are compact. As Shapiro and Sundberg [14] indicate in their paper, "compact composition operators are dramatically nonisolated." They show that the set of compact composition operators is path connected, and therefore these operators are never isolated. It is interesting to ask which composition operators are, in fact, isolated. Shapiro and Sundberg studied this problem, and showed (among other things) that if ¢ is an analytic self-map of D that is not an extreme point of the algebra H 00 (D), then Cq, is not isolated; in their words, "isolated composition operators can only be induced by extreme points." This allowed them to exhibit an example of a non-compact non-isolated operator. They also raised several questions at the end of their paper: (1) Characterize the components in Comp(H 2 ), the space of all composition operators on H 2 . (2) Which composition operators are isolated? (3) Characterize composition operators whose difference is compact. Before stating the final question, we remind the reader that the essential norm of an operator T defined on a Banach space H is the distance to the compact operators; that is,

IITIIe = inf{))T- K)) : K compact on H}. It is clear that ))T)) ~ ))T))e, and therefore every essentially isolated operator is isolated. In fact, because of the abundance of weakly null sequences in H 2 , all the results on isolation appearing in Shapiro and Sundberg's paper hold true if we replace the norm with the essential norm (see [14], p. 148). Thus they raised the following question.

(4) Is every isolated operator essentially isolated? Other papers of interest on this subject include [10]. Of course, one is not limited to the space H 2 , and the study of composition operators on various spaces has lead to a large body of literature. In this paper, we are interested in the same problems for composition operators on H 00 ( Bn)' where Bn denotes the open unit ball in en. While the problem on H 2 seems to be difficult, MacCluer, Ohno and Zhao [11] were able to obtain partial results about operators on the algebra H 00 (D). They showed that for two analytic self-maps of the disk, Cq, and C1/J are in the same path component in the space of composition operators, Comp(H 00 (D)), if and only if ))Cq,- C1f;)) < 2. In particular, an operator Cq, is isolated in Comp(H 00 (D)) if and only if ))Cq,- C1f;)) = 2 for any other analytic self-map '1/J of the disk. The authors show that this result can be rephrased in terms of the pseudohyperbolic metric, and they posed the question of whether or not every isolated composition operator on H 00 is, in fact, essentially isolated. The answer to this last question was given by Hosokawa, Izuchi and Zheng [8]. Their technique was to develop something called asymptotic interpolating sequences, or a.i.s. for short. Essentially, this definition allowed them to interpolate sequences with a good bound on the norm (see [6] for more information about these sequences). They then used these sequences and Blaschke products to obtain H 00 (D) functions that provide good estimates for the essential norm of the difference of two composition operators.


HOMOTOPIC COMPOSITION OPERATORS ON H=(B")

179

In this paper, we give simpler proofs of the results obtained by Hosokawa, Izuchi and Zheng, and combine them with the proofs of MacCluer, Ohno and Zhao. Because our proofs are significantly simpler and do not refer to asymptotic interpolating sequences or Blaschke products, we are able to obtain the results on the ball in en. While our results do not rely on interpolating sequence results, they do rely on a construction of Gamelin and Garnett relating interpolating sequences to peak sets [5]. The proof we will provide is simple enough to be applicable to other algebras. We conclude the paper with an example of such an application: Using these same techniques, we are able to "lift" these results to obtain a characterization of the path components of endomorphisms of H 00 (D). After we completed this paper, we learned that some of the results on composition operators on H 00 (Bn) were also obtained by Carl Toews [15]. Two other papers directly related to the results described here are [3] and [9]. Finally, we mention that recent results on Shapiro and Sundberg's first question in the space H 2 can be found in [12]. 2. Preliminary results

Our goal is to prove results about composition operators on H 00 ( Bn) and endomorphisms of H 00 (D). In this section, we present proofs of several lemmas that will be important in obtaining estimates on norms and essential norms of operators. Our discussion begins with functions of n variables and z = (z 1 , z 2 , ... , zn), where each z 1 is a complex number. As usual, the associated norm of z is given by

and the unit ball Bn is the set of all z E en for which lzl < 1. We let H 00 (En) denote the space of bounded holomorphic functions on Bn. If n = 1 our situation reduces to the familiar space of functions on the unit disk: H 00 (D). We need some background on general uniform algebras, some information specific to H 00 ( Bn) and some deeper results for H 00 (D). Everything we need is presented in this paper. Let A be a uniform algebra. The maximal ideal space of A, denoted M(A), is the set of complex-valued, linear, multiplicative maps of A that map the identity of A to the value 1 E C. Since evaluations at points of Bn are linear multiplicative functionals on H 00 (Bn), we may think of Bn as a subset of M(H 00 (Bn)). It is well known that M(A) is a compact Hausdorff space when endowed with the weak-* topology induced by A*. We will always consider this topology for M(A). For an element a E A, the Gelfand transform of a is a complex-valued map defined on M(A) by ii(x) = x(a). This map establishes an isometric isomorphism between A and a closed subalgebra of C(M(A)). It is usual to identify the function with its Gelfand transform, since the meaning is generally clear from the context. For x, y E M(A), the pseudohyperbolic and hyperbolic metrics are defined, respectively, by

p(x, y) = sup{lf(y)l : f

E

A, 11!11 :::; 1, and f(x) = 0} and

h( x, y ) -- log 1+p(x,y) ( ). 1- p x, y


180


It is well-known that pis a [0, 1]-valued metric and that h is a [0, +oo]-valued metric on M(A). The triangle inequality for h immediately implies that the condition p(x,y) < 1 (i.e. h(x,y) < oo) is an equivalence relation on M(A). If A is a uniform algebra and¢: M(A)-+M(A) is a continuous map such that a o ¢ E A for all a E A, then the map C¢ defined by

Ccf>a =a o ¢ is an endomorphism of A. We will write C¢ E End (A). Conversely, if E E End (A), we may define¢: M(A)-+M(A) by cj>(x)(a) = x(E(a)), obtaining E = Ccf>. The Shilov boundary of A is the smallest closed subset of M(A) on which every function in A attains its maximum. We denote the Shilov boundary by 8A. LEMMA 2. Let A be a uniform algebra and let Ccf>, C'l/1 E End (A). Let V C M(A) be a set whose closure contains 8A. Then,

(2.1)

sup xEV

PROOF.

(2.2)

2p(¢(x), 'tj;(x)) ::::; IIC¢- C'l/111 ::::; 2 sup p(cp(x), 'l/J(x)). 1 + J1- p(cp(x), 'tj;(x)) 2 xEV

First we show that if x, y

E

M(A), then

2p(x,y) ::s;llx-yiiA·:::;2p(x,y). 1+J1-p(x,y) 2

The proof uses the techniques of [4, p. 144]. Let f E A with 11!11 < 1. It is clear that J = (!- f(y))/(1- f(y)f) E A, J(y) = 0 and llfll ::::; 1. By definition of p then lf(x)l ::::; p(x, y), and consequently

lf(x)- f(y)l ::::; 11- f(y)f(x)l p(x, y) ::::; 2p(x, y). Taking the supremum over 11!11 < 1 we get the upper inequality. Now we turn to the lower inequality. For simplicity, we write p = p(x, y). Choose fn E A with llfnll < 1, fn(x) = 0 and fn(Y) > p-1/n. Let Pn = fn(Y) and Ln(z) = (tn- z)/(1- tnz), where tn = (1- J1- p~,)/ Pn· Therefore Ln o fn E A, IILn o fnll ::::; 1 and llx- YIIA• 2: I(Ln

0

fn)(x)- (Ln o fn)(y)l = ltn- Ln(Pn)l.

A simple computation shows that ltn- Ln(Pn)l

=

IPn( 1 -

2p t;) ,___. 1-tnPn 1+~

•

The lemma will follow immediately from (2.2) and the following chain of identities: sup sup I(! o cp)(x)- (! o 'tj;)(x)l 11!11=1 xE8A sup sup I(! o cp)(x)- (! o 'tj;)(x)l 11!11=1 xEV sup sup lf(cp(x))- f('tj;(x))l xEV 11!11=1 sup ll¢(x)- 'l/J(x)IIA·· xEV

D LEMMA 3. Let A be a uniform algebra and let Ccf>, C'l/1 liCe/>- C'l/111 < 2 is an equivalence relation.

E

End (A). The condition


HOMOTOPIC COMPOSITION OPERATORS ON

181

H 00 (Bn)

PROOF. It is obvious that the relation is reflexive and symmetric. So, suppose that ¢, '1/J and cp define endomorphisms of A such that IICq,- C,p I < 2 and IIC,pC'PII < 2. By (2.1) we see that

a 1 = sup p(¢(x), 'lj;(x)) < 1 and a2 xEM(A)

=

sup p('l/J(x), cp(x)) < 1.

xEM(A)

Using the hyperbolic metric h on M(A) we obtain 1 + a1 1 + a2 h(¢(x), cp(x)):::; log -1 - - +log -1 - xEM(A) - a1 - a2

sup

=

/3,

and consequently supxEM(A) p(¢(x), cp(x)):::; (e/3 -1)/(e/3 + 1) < 1. A new application of (2.1) yields the desired result. D LEMMA 4. Let A be a uniform algebra. If {fn} is a sequence of functions in the unit ball of A tending pointwise to zero on 8A, then Un} tends to zero weakly in A. PROOF. As indicated in the introduction, using the Gelfand transform, we may think of A ~ C(8A). Let x be any element of the dual space of A. By the Hahn-Banach theorem, x has a continuous norm preserving extension to the space of continuous functions on the Shilov boundary. Therefore, there exists a finite measure llx on 8A such that

x(f)

=

rf laA

dl"x'

But llfnll :::; 1 for all n, and fn ----+ 0 pointwise on 8A, so we may apply the Lebesgue dominated convergence theorem to conclude that x(fn) ----+ 0. Therefore, the sequence {fn} converges to zero weakly. D We will need another estimate, but this will depend on the pseudohyperbolic metric particular to the ball, Bn. For a, z E Bn, let sa = )1 - lal 2 , and the and Qa = I - Pa. Relevant a, a computations can be found in [13, p. 25]. On Bn, the pseudohyperbolic metric induced by H 00 (Bn) is given by projections Pa and Qa be given by Pa(z)

(

p a, z

= ((z, a)) a

) =la-Pa(z)-saQa(z)l 1 - (z, a) '

In what follows, for points a and z in the ball and numbers s and t in the closed interval [0, 1], we let a 8 =a+ s(z- a) and as+t =a+ (s + t)(z- a). LEMMA 5. Let a, z E Bn. For s, t E [0, 1] satisfying (2.3)

p(a+s(z-a),a+(s+t)(z-a)):::;

t:::;

1- s we have

/p(a,)z)(

1- 1- t p a, z

)

PROOF. We can assume that a =1- z, because otherwise there is nothing to prove. We consider first the case s = 0. Since Pa and Qa are linear operators satisfying Pa(a) =a and Qa(a) = 0, the numerator of p(a, a+ t(z- a)) satisfies a- Pa(a

+ t(z- a))- saQa(a + t(z- a))= ta- tPa(z)- tsaQa(z).


182


We have assumed that a -/= z, and therefore a- Pa(z) - saQa(z) -/= 0. A simple computation gives

p(a, a+ t(z- a))

a- Pa(a + t(z- a))- SatQa(z) I 1 - (a+ t(z- a), a) t(a- Pa(z)- saQa(z)) I I1 - (z, a) + (z, a) - (a, a) - t(z - a, a)

I

< 11/ p(z, a) - 1(1 - t)(z- a,ta) 1/la- Pa(z) - saQa(z)ll· But l(z- a, a) I= l(a- Pa(z)- saQa(z), a) I :S: Ia- Pa(z)- saQa(z)llal. Therefore

(2.4)

t p(a, at) :::;: (1/ p(a, z))- (1- t)

tp(a,z)

1- (1- t)p(a, z) ·

This proves (2.3) for s = 0. For the general case we can assume s -/= 1 since otherwise t = 0 and there is nothing to prove. From (2.4) we obtain

p(as, as+ t(z- a)) p(as,as + (t/(1- s))(z- as))

: H 00 (Bn) ___, H 00 (Bn) defined by C¢(!) = f o ¢. These maps are all endomorphisms of the algebra H 00 ( Bn). For the special case of n = 1 we will say more in the final section of the paper. We are interested here in estimates on the essential norm of the difference of two composition operators. If T is a bounded operator, we denote its essential norm by IITIIe· THEOREM

6. Let¢ and 'ljJ be holomorphic self-maps of Bn such that max{ll¢11, 11'1/JII} = 1.


HOMOTOPIC COMPOSITION OPERATORS ON H 00 (Bn)

183

Let

e=

max {lim sup p( ¢(z), 1/J(z) ), lim sup p( ¢(z ), 1/J(z))} . l(z)J-->1

11/>(z)J-->1

Then

(3.1) PROOF. By hypothesis there is a sequence of points { Zj} in En, such that p( ¢( Zj), 1/J( Zj)) ---> e, and one of them, say {¢( Zj)}, converges to a point ( on the boundary of En. Without loss of generality we may assume that p(¢(zj), 1/J(zj)) > e-1/j. Let kj E H 00 (En) be a function of norm one, whose existence is guaranteed by the pseudohyperbolic distance definition, satisfying kj(¢(zj)) > e- 1/j and kj (1/J(zj)) = 0. Consider the functions

f(z)=(1+(z,())/2 and g(z)=(1-(z,())/2. Then J,g E H 00 (En), /(() = 1, lf(1J)I < 1 for all17 E 8En satisfying 1J-=/:- (,and g(() = 0. We will now produce a sequence of functions, {hj}, tending to zero weakly for which lhj(¢(zj))- hj('lj;(zj))l---> (!. We proceed as follows. Let j E N. Since f(¢(zj)) ---> 1, we may choose Zm 1 so that lf(¢(zm1 ))1i > J1- 1/j. Now since g -=1- 0 on En, there exists an integer lj such that lg(¢(zm1 ))1 1 / 11 ~ yf1- 1/j. Consider the functions hj = (ll 11)(fj)kmr Then llhjll:::; 1, hj('l/J(zm1)) = 0, and lhj(¢(zm1 ))1 > (1- 1/j)(e- 1/mj)· We note that hj---> 0 on the Shilov boundary and, by Lemma 4, hj ---> 0 weakly. Thus for any compact operator K we have IICq,- C,p + Kll

> IICq,hj- C,phj + Khjll ~

=

lhj(¢(zmJ)- hj('l/J(zm1)) + (Khj)(zm 1)1 lhj(¢(zmJ) + (Khj)(zmJI---> (!.

This proves the lower inequality in (3.1). For the upper inequality, let c > 0 and choose 8 with 0 < 8 < 1 close enough to 1 so that p(¢(z), 1/J(z)) :::;

e+ c

on the set {l¢(z)l > 8} U {11/J(z)l > 8}.

Now choose a= a(c, 8) E (0, 1) close enough to one so that p(¢(z), a¢(z)) < c and p('l/J(z), a'lj;(z)) 1

= lim sup

j,P(z)j->1

p(q)(z), 'ljl(z))

= 0.

PROOF. It is clear that Cq., and C,p are compact, if max{ll¢11, ii'ljlii} < 1. On the other hand, if max {11¢11, li'ljlii} = 1 and e is the parameter of Theorem 6, then (3.1) says that Cq.,- C,p is compact if and only if e = 0. 0 Our next goal is to characterize the path components of composition operators on H 00 (Bn). We write Cq., rv C,p to indicate that there is a norm-continuous homotopy of composition operators joining Cq., with C,p. Also, if K denotes the ideal of compact operators, we write Cq., "-'e C,p to indicate that there is an essential norm-continuous homotopy of classes {C'P + K : 'P : Bn--. Bn holomorphic} joining Cq., + K with C,p + K. Let q) be a holomorphic self-map of Bn. For x E M ( H 00 ( Bn)) we can define q)(x) E M(H 00 (Bn)) by the rule q)(x)(f) ~f x(f o ¢). Thus we can extend q) : Bn-.Bn to a self-map of M(H 00 (Bn)), which we also denote by¢. The continuity of this extension is immediate. We now have everything we need to prove the main theorem of this paper. As indicated in the introduction, this theorem unifies and extends many of the results appearing in [11], as well as [8].

THEOREM 8. Let q) and 'ljJ be holomorphic self-maps of the unit ball in Then the following are equivalent. (a) Cq.,

en.

C,p. (b) Cq., "-'e C,p. (c) IICq.,- C,pll < 2. (d) SUPzEBn p(q)(z), 'ljl(z)) < 1. rv

PROOF. (a) =} (b) is obvious. (c) 9 (d). A boundary for H 00 (Bn) is a closed set F C M(H 00 (Bn)) such that IIJII = supxEF lf(x)l for all f E H 00 (Bn). It is clear that the closure Bn of Bn in M(H=(Bn)) is a boundary for H=(Bn), and since oH=(Bn) is the intersection of all the boundaries [4, p. 10], then oH= (En) c Bn. The equivalence then follows from (2.1). (b) =} (c). By hypothesis there is a family { 0 such that for z, wE (3D with lz- wl < 8 we have lf(z)- f(w)l

sup l(ACL,(f o bk))(z)l

zErD

n

sup IL>..j(fobkoLx)(¢j(z))l zErD

>

j=l n

sup I L

Ajf(¢j(z))l- ME

j=l > (1- E) 2 IIAII- ME.

Letting

f

--->

zErD

0 yields the desired result.

D

THEOREM 11. Let T1 , T 2 E End (H 00 (D)). Then the following are equivalent.

(a) T1 "'T2 in End(H 00 (D)). (b) IITt- T2il < 2. (c) There exist x E M(H 00 (D)) and holomorphic self-maps ¢, 'ljJ of D such that T1 = Cq;CL,, T2 = C.pCL, and IJCq;- C.pll < 2. PROOF. Suppose that (a) holds. Then there is a homotopy

G: [0, 1] ___,End (H 00 (D)) with G(O)

=

T 1 and G(1)

tn = 1 such that

= T2 .

We can find finitely many points 0 = h < ... < 2 for j = 1, ... , n- 1. Lemma 3 then says

IIG(tj)- G(tJ+l)ll
(Md,1 U Md,-d· (The set is finite by Theorem 2.3.) Then for every a E {a E A: a(xj) = O,j = 1, 2, ... , n} p(a)(y)

=

{Ty(a(q>(y))), 0,

y E M1 U M_ 1, y E Mo U Md,1 U Md,-1·

Since Ty is real-linear for every y E M 1 U M_ 1 , the conclusion follows.

D

COROLLARY 3.3. Let p be a ring homomorphism from A into B such that p(CeA) = Ces. Then we have that M 0 = 0, and there exists a continuous mapping q, from Ms into MA such that one of the following three occurs. (1) p is linear: p(a)(y)

= a(q>(y)),

a E A,

y EMs.

a E A,

y EMs.

(2) p is conjugate linear: p(a)(y)

= a(q>(y)),

(3) There exists a non-trivial ring automorphism T on C such that p(a)(y)

= T(a(q>(y))),

a E A,

y EMs.

In particular, if there exists an a E A such that the spectrum of p(a) is an infinite set, then p is linear or conjugate linear.

PROOF. For every y EMs, we have Py(CeA) = C, so Py(A) = Py(CeA) =C. Thus ker Py is a maximal ideal of A by Lemma 2.2, so that the condition (m) is satisfied. Since p(ieA) E Ces, Ms = M1 U Md,1 or Ms = M-1 U Md,-1· Suppose that Md, 1 U Md,- 1 = 0. Then (1) or (2) occurs. Suppose that there exists some Yd E Md,1 and some Y1 E M1. Then there is a complex number A with Tyd(A) #A, so that p(AeA) is not a constant function, which contradicts our hypothesis. Thus Md, 1 # 0 implies that Ms = Md,1· It is also easy to see that Ty is identical for every y E Md, 1 since p(CeA) consists of constant functions. Thus (3) follows. In the same way we see that (3) follows if Md,- 1 # 0. Suppose that there exists an a E A such that p(a)(Ms) is infinite, then (3) does not occur since q>(Ms) is a finite set in this case. It follows that pis linear or conjugate linear. D


AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS

209

Corresponding results for ring homomorphisms on rings of analytic functions are proved by Kra [10, Theorem I]. Suppose that p is a ring homomorphism from A into B which satisfies two conditions: p( q. Our second main topic is to characterize when I is order bounded, that is, the unit ball of A~ is a subset of some order interval in Lq(f..l). As a consequence, we obtain necessary and sufficient conditions for I to have specific (absolutely) summing properties. Our results extend corresponding ones for composition operators which have been obtained e.g. in [3], [4], [20], [21]. In fact, they can also be viewed as results on composition operators which may have rather unusual range spaces. They apply, for example, to pointwise multipliers. Many of the results to be presented remain valid for measures f..l on D for which f f---+ f induces just a bounded linear map A~ --+ U(f..l) (not necessarily injective). This is rather straightforward; precise formulations, however, require somewhat 1991 Mathematics Subject Classification. Primary 46 E 15, 47 B 38, 47 B 10; Secondary 46 B 25, 30 H 05, 32 H 10. Key words and phrases. Weighted Bergman spaces, Carleson measures, composition operators, compactness, order boundedness, absolutely summing operators. The results of this paper are part of the dissertation of the second named author written at the University of Ziirich under the supervision of the first.

217

©



218

HANS JARCHOW AND URS KOLLBRUNNER

clumsy notation. Also, there is an immediate extension to complex Borel measures on llJ whose variation is (a, p, q)- Carleson. We are indebted to the referee for providing Example 8 and for bringing to our attention the paper [15] by V.L. Oleinikov and B.S. Pavlov.

2. Weighted Bergman spaces Throughout the paper, we will use standard results and notation from (quasi- ) Banach space theory. We will work on the open unit disk llJ = {z E C: lzl < 1} in the complex plane. The space 1t(ll.J) of all analytic functions llJ---> C is a Frechet space with respect to the topology of uniform convergence on compact subsets of ll.J. Let d(J be normalized area measure on ll.J. For each a > -1,

d(Ja(z)

:=

(a+ 1) (1

is a probability measure on ll.J. For each 0 Bergman space is defined to be A~

:=

-lzl 2 )"" d(J(z)

< p < oo, the corresponding weighted

1t(ll.J) n £P((Ja)·

is closed in £P ((J""); it is a Banach space if p ~ 1 and a p- Banach space if 0 < p < 1. Its (p-) norm will be denoted by II ·lla,p· A~ is a Hilbert space and has a reproducing kernel:

A~

Ka(z, w) = K(z, w)""+ 2 ; here K ( z, w) = ( 1 - zw) - 1 is the reproducing kernel for the Hardy space H 2 • For reasons like this, the scale of Hardy spaces is often considered as the scale of weighted Bergman spaces which corresponds to a = -1. Some of the results below actually remain true for this case, and some can even be extended to analytic Besov spaces 13~ (f E 13~ {:} f' E A~+p)· Nevertheless, in this paper we will only deal with the case -1 < a < oo.

3. Carleson measures All measures on llJ will be finite, positive Borel measures. Let -1 < a < oo and 0 < p, q < oo be given. We say that a measure J.L on llJ is an (a,p, q)- Carleson measure if A~ C U(J.L) and the embedding A~ 0 such that IIJIILq(J.L) :S: C ·IIJIIA~ ' Lq(J.L) will be referred to as a Carleson embedding. As mentioned in the introduction, a number of the results to follow remain true if we just require that f r-+ f induces a bounded linear map A~ ---> Lq(J.L). Also, complex measures whose variation is (a, p, q)- Carleson can be incorporated. Moreover, there are extensions to analytic functions of several variables. However, we are not going to discuss such generalizations in this paper. We say that J.L is a compact (a,p, q)- Carleson measure if the embedding A~ Lq((J(3) : f r-+ fh iff the measure hq d(Jf3 is (a,p, q)- Carleson. Moreover, discrete (a,p, q)- Carleson measures on llJ can be defined using appropriate versions of 'sampling sequences', etc.


CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES

219

An important example is obtained by looking at the composition operator C'P : f o ({1 induced by a non-constant analytic function ({1 : liJ ----> liJ. Clearly, C'P :A~----. A~ exists iff a a o ({J- 1 is (a,p, q)- Carleson. More generally, an arbitrary measure f.l on liJ is (a,p, q)- Carleson if and only if, for every analytic map ({1: liJ----> liJ, C'P maps A~ boundedly into Aq(f.l) := 'H(liJ) n Lq(f.l). In fact, the condition applied to the identity ofliJ shows that 1-l is (a,p,q)Carleson. On the other hand, if 1-l is (a, p, q)- Carleson and ({1 : liJ ----> liJ is analytic, then C'P :A~ ----. Aq(f.l) is well-defined and bounded. For non-constant functions ({J, the condition is further equivalent to 1-l o ({J- 1 being (a,p, q)- Carleson. This allows an interpretation of Carleson embeddings, and in particular of multipliers as above, as composition operators. However, in such a general setting the range space of a composition operator might be unpleasent, and desirable properties may not be available. For example, Aq(f.l) embeds continuously into 'H(liJ) if and only if Aq(f.l) is a closed subspace of Lq(f.l) and all point evaluations Aq(f.l) ----> C : f r-+ f(z), z E lU, are continuous. (a, p, q)- Carleson measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking

f

r-+

[11],[12].

.1

The hyperbolic metric on liJ is given by Q(z, w) := 1~f

'Y

ld(l

1 -l(l

where the infimum extends over all smooth curves "( in liJ joining z and w. For w E liJ arid r > 0, let Br(w) = {z E liJ: Q(z,w) < r}

be the corresponding hyperbolic disk. Actually, the particular choice of r > 0 doesn't really matter in our context. Let us agree to write A~'-+ Lq(f.l) if A~ is a subset of Lq(f.l) and the embedding is continuous. Similar for other function spaces. The Carleson measures under consideration can be characterized in terms of the function

THEOREM

3.1. Let -1 0 such that 1 C · aa(Br(w)) :::; (1 -lwl 2) 0 +2 :::; C · aa(Br(w))

'Vw E lU.

Therefore we may also say that Theorem 3.1 refers to properties of the function

w

r-+

f.l(Br(w))1fq(1 -lwl2)-(a+2)/P .


220


It also follows that Ha,p,q E Lf!q (A) if and only if f..L(Br(·))/(J"a(Br(-)) is in LP/(p-q)((J"a)· This will be used in the proof of Theorem 4.3 below. If p :::; q, then the relevant parameter in Theorem 3.1 is q (a+ 2)/p, whereas for p > q and fixed a, dependence is on pjq. As a first immediate consequence we may state: COROLLARY

if and only if A~

3.2. For any -1 -1 (see C.

3.5. Suppose that -1 < a, (3 < oo andO < p, q < oo. iff (a+ 2)/p:::; ((3 + 2)/q. (a) If p:::; q, then A~ '--+A~

COROLLARY

(b) If p > q, then A~'--+

iff (a+ 1)/p < ((3

A~

+ 1)/q.

There are several ways to modify the domain space of a composition operator. In a systematic fashion, we may proceed as follows; cf. [4]. Each of the kernel functions Ka(z, ·) is bounded (z E 1U), and k

has (p-) norm one in representation

A~

w ·= Oi,p,z( ) .

(

1-

lzl2 ) (a+2)/P

(1- zw)2

(0 < p < oo). The functions

f

E A~

which admit a

n=l

where the scalars an satisfy Ln linear space, say

lanl
1

' '

PROOF. (i) =? (ii): If p 2 1 then nothing is to prove since A~) '-----' A~. If p < 1 :::; q then, by [19], A~) is the Banach envelope of A~, and the convex hull of BAr, is dense in B A~>· Hence relative compactness of BAr, in the Banach space Lq(fJ) entails relative compactness of B A~l in Lq(fJ). -Here we have used Bx to denote the unit ball of a (quasi-) Banach space X. (ii) =?(iii): Suppose that (iii) doesn't hold. Then there exist an E > 0 and a sequence (zn) in 1U such that limn_,= lznl = 1 and llka,p,zn IILq(M) > E for all n. By (ii), we may assume that (ka,p,zn)n converges to some f E Lq(fJ). But limn_,= lznl = 1 implies f = 0 since clearly (ka,p,zJ tends to zero pointwise: contradiction. (iii)=? (iv): The estimate proved in (iii)=? (i) of Proposition 3.6 provides us with a constant C = C(r) > 0 such that fJ(Br(w)) 1 fq :::; C · (1- lwl 2 )(a+ 2 )/P · llka,p,wiiLq(M) for all WE 1U.



223

(iv) ==? (i): We apply Lemma 4.3.6 of K. Zhu [22]: there exists an integer N such that for sufficiently small r there is a sequence (TJn) in 1U having the following properties:

(1) 1U = U~=l

Br(TJn), (2) Br;4(TJm) n Br;4(TJn) = 0 whenever m -j. n, (3) Every z E 1U is contained in at most N of the sets B2r(TJn)· Note that limn_, 00 ITJnl = 1 follows from (2). Let now Un) be a sequence in BA'/,· By Montel's theorem, some subsequence Unk) converges uniformly on compact sets to some f E "H(lU). By Fatou's lemma, even f E BA'/,· Put gk := f- fnk, 'Vk E N. By our hypothesis there exists, for any given c > 0, an integer ne: such that p(Br(TJn)) ~ c · (1- l77nl 2 )q(a+ 2)/p if n ;::: nE. Therefore, with constants depending only on the indicated parameters,

< C(r). C(r,o:) · Nqfp. c ·

(l

lgk(w)IPdCJa(w)r 1p

~ c · c;

here C = C(r,o:,N,p,q). If we choose now kE EN such that I::n Lq(p) if -1 q, we can prove: THEOREM 4.3. Suppose that -1 L 1 (J.L) is compact. Since A~ is reflexive it suffices to verify that I is completely continuous. Accordingly, let Un)n be a weakly null sequence in A~. Then fn(z)-+ 0 for each z E 1I.J. Being weakly null in L 1 (J.L), Un) is uniformly integrable. llfnlh = 0, by a theorem of Vitali (see W. Rudin [18], p.133). Now limn~oo Standard results from interpolation theory on the preservation of compactness by interpolated operators lead from either of these special results to (at least parts of) Theorem 4.3. 5. Order bounded and absolutely summing operators

Our Banach lattices will be complex Banach lattices; see e.g. P. Meyer-Nieberg

[13] for the construction of such an object from a real Banach lattice. Let X

be a Banach space and Y a closed subspace of a Banach lattice L. An operator u : X -+ Y is called order bounded if there is a non- negative h E L such that lufl ::; h for fin Bx, the unit ball of X. Thus we require u to map Bx into the order interval {g E L : lgl ::; h} of L. Note that L is part of the definition! Every u E £(X, Y) is order bounded when Y is considered as a subspace of C(K) for some compact Hausdorff space K. Let I be an order interval in the Banach lattice L. Its span, Z, is a Banach lattice with respect to L's order and (a multiple of) I's gauge functional as its norm. Z is an abstract M- space with unit and so, by a well-known theorem of S. Kakutani, isometrically isomorphic (as a Banach lattice) to C(K) for some compact Hausdorff space K; see again [13]. It follows that every order bounded operator u : X -+ Y C L factorizes X ~

Z ~ C(K) J_. L where K is as before and j is the canonical embedding. In this paper, L will be a space £P(J.L) which results in close ties with absolutely summing operators. Recall that a Banach space operator u : X -+ Y is ( q,p) summing (p::; q), written u E IIq,p(X, Y), if there is a constant C such that, for every choice of n EN and x 1 , ... , Xn EX, n

(LIIuxkllq) k=1

1/ q

::;

C· .sup x EBx•

n

(LI(x*,xkW)

1/ P

k=1

In other words, u is (q, p)- summing iff every weak f!P- sequence, i.e. every sequence (xn) in X which satisfies 2:~= 1 l(x*, Xn) IP < oo for all x* E X*, is taken to a strong Rq- sequence, i.e. 2:~= 1 lluxn llq < oo holds. (p, p)- summing operators are called p- summing; the corresponding notation is IIp(X, Y) = IIp,p(X, Y). We refer to [2] for details on these concepts and in particular for the following facts: • If H and K are Hilbert spaces and q 2 2, then IIq, 2 (H, K) is the corresponding Schatten q- class. Moreover, for any 1 ::; p < oo, IIp(H, K) is the class of Hilbert- Schmidt operators. • If 1 ::; p::; 2, then every operator from C(K) to £P(v) is 2- summing.



226

• If p > 2, then every operator C(K) ----+ LP(v) is (p, 2) -summing, and r- summing for every r > p.

Moreover: • If u: X----+ Lp(v) is order bounded then u is p- summing.

Here v is any measure. In the last statement, the converse fails. But: • If u* p- summing then u is order bounded.

More precisely, we have the following result due to D.J.H. Garling [5]: • Let 1 :S p < oo. A Banach space operator u : X ----+ Y has a p- summing adjoint if and only if, for every measure v and operator v : Y ----+ LP (v), the composition v o u: X----+ LP(v) is order bounded.

In particular: • An operator u : L 2 (vi) ----+ L 2 (v2 ) is order bounded iff it is HilbertSchmidt.

We are going to characterize order boundedness of Carleson embeddings

Lq (;.t). To this end we introduce, for s > 0, the Banach space

Xs

:=

{f: 1U----+ C:

f

measurable, sup(1zE1U

and its closed subspace

Xs := Xs It is easy to see that A~'---+

n

Jzj 2 ) Jf(z)J < oo} 8

A~

'---+

(ii) A~)'---+

'---+

.

1-l(1U) .

X(a+2)/p and that the index (a+2)/p is best possible.

THEOREM 5.1. Let -1

Q(rJn)

n

lvn(zW dp,(z)

( L1

L1

1lJ

n

(1 -11],~1/p*) 11 -1]nZI

Q(rJn)

n

> C.

n

q(a+2)

(1 -lzl2)-q(a+2)/Pdp,(z) .

Q(ryn)

Thus (1 -lzl 2)-(+ 2l/P E Lq(p,), and so I is order bounded by Theorem 5.1.

0

More is available. Consider the Rademacher functions rn: [0, 1]

-----+

ffi. : t

f---+

sign sin (2n7rt) , n EN

(or any sequence of independent symmetric Bernoulli variables). Given 0 < p < oo, Khinchin's inequality assures the existence of positive constants Ap and Bp such that, for any finite collection of scalars a 1 , ... , an: Ap.

f

(~

laki 2 12 :::;

(fo 1 ~akrk(t)lp 1

1 dtf p:::; Bp.

(~

f

laki 2 12 .

Pursuing the fate of this inequality within the framework of Banach spaces leads to the theory of type and cotype of Banach spaces, and to the following related class of operators (compare [2], Chs. 11-12). A Banach space operator u : X ----+ Y is almost summing, u E IIas(X, Y) ,

if there is a constant C such that, for any choice of finitely many vectors x 1 , ... , Xn from X,

(jQ{1

11

n

2 )1/2

t;rk(t)UXk 11 dt

:::;

cx*~~X*

(

n )1/2 t;l(x*,Xk)l 2

It is known that each of the operator ideals IIp is properly contained in IIas. Moreover, if 1 :::; p < oo and r = max{p, 2} then IIas(·, X) C IIr,2(·, X) whenever X is an J:P space, or the Schatten p-class Sp(H) for some Hilbert space H. In addition, it was shown by S. Kwapieri [9] that • if His a Hilbert space and u is in IIas(H, Y) then the adjoint u* : Y* ----+ H is 1- summing. See [2], p.255 for details. We have the following application to Carleson embeddings. The argument is the same as for composition operators between weighted Bergman spaces [3]. PROPOSITION 5.6. Let p, be an (a,p,q)- Carleson measure where 1:::; q < oo, and 2 :::; p < oo. The embedding I: A~ '----+ Lq(p,) is almost summing if and only if it is order bounded. PROOF. Definer> -1 by (r + 2)/2 =(a+ 2)/p. Since p 2 2' A;'----+ A~. =? I : '----+ Lq (p,) is almost summing =? I* is 1 - summing (K wapieri) =? I is order bounded (Garling)

A;



229

Combining the preceding two propositions yields: COROLLARY 5.7. Let -1 < o: < oo and 1 ::::; p,q < oo be such that p ~ min {q*, 2} and let p, be an (o:, p, q) - Carles on measure. The embedding I : A~ ~

Lq (p,) is q -summing iff it is order bounded.

PROOF. Only sufficiency requires proof. If p ~ q*, then Proposition 5.5 settles the case. And if p ~ 2, then I, being q- summing, is almost summing, and so order bounded by Proposition 5.6. D

References [1] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103. (1976), 611-635. [2] J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators. Cambridge University Press 1995. [3] T. Domenig, Composition operators on weighted Bergman spaces and Hardy spaces. Dissertation University of Ziirich 1997. [4] T. Domenig, H. Jarchow, R. Riedl, The domain space of an analytic composition operator. Journ. Austral. Math. Soc. 66 (1999), 56-65. [5] D.J.H. Garling, Lattice bounding, Radonifying and summing mappings. Math. Proc. Camb. Phil. Soc. 77 (1975), 327-333. [6] W.W. Hastings, A Carleson measure theorem for Bergman spaces. Proc. Amer. Math. Soc. 52 (1975), 237-241. [7] C. Horowitz Zeros of functions in the Bergman spaces. Duke Math. Journ. 41 (1974), 693-710. [8] S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing operators. Studia Math. 38 (1970), 193-201. [9] S. Kwapien, A remark on p- summing operators in lr- spaces. Studia Math. 34 (1970), 277278. [10] J. Lindenstrauss, A. Pelczynski, Contributions to the theory of classical Banach spaces. Journ. Funct. Anal. 8 (1971), 225-249. [11] D.H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces. Proc. Edinb. Math. Soc. 29 (1986), 125-131. [12] D.H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine 's inequality. Mich. Math. Journ. 40 (1993), 333-358. [13] P. Meyer-Nieberg, Banach Lattices. Springer-Verlag 1991. [14] E. Oja, Pitt Theorem for non-locally convex spaces £p. Preprint. [15] V.L. Oleinik, B.S. Pavlov, Embedding theorems for weighted classes of harmonic functions. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 22 (1971), 94-102. Trans!. in Journ. Soviet Math. 2 (1974), 135-142. [16] H.R.Pitt, A note on bilinear forms. Journ. London Math. Soc. 11, 171-174 (1936). [17] H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces with an appendix on compactness of operators from LP(J.L) to Lr(v). Journ. Funct. Anal. 4 (1969), 176-214. [18] W. Rudin, Real and Complex Analysis. 3rd ed., McGraw-Hill 1987. [19] J.H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces. Duke Math. Journ. 43 (1976), 187-202. [20] W. Smith, Composition operators between Bergman and Hardy spaces. Trans. Amer. Math. Soc. 348 (1996) 2331-2348. [21] W. Smith, L. Yang, Composition operators that improve integrability on weighted Bergman spaces. Proc. Amer. Math. Soc. 126 (1998) 411-420. [22] K. Zhu, Operator Theory in Function Spaces. Marcel Dekker, New York 1990.


230

HANS JARCHOW AND URS KOLLBRUNNER lNSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE

190, CH 8057

ZURICH, SWITZERLAND

E-mail address: j archow@math. unizh. ch lNSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE ZURICH, SWITZERLAND

E-mail address: kollbrun@math. unizh. ch


190, CH 8057

http://dx.doi.org/10.1090/conm/328/05784 Contemporary Mathematics

Volume 328, 2003

Weak*-extreme points of injective tensor product spaces Krzysztof Jarosz and T. S. S. R. K. Rao We investigate weak* -extreme points of the injective tensor product spaces of the form A@, E, where A is a closed subspace of C (X) and E is a Banach space. We show that if x E X is a weak peak point of A then f (x) is a weak*-extreme point for any weak*-extreme point fin the unit ball of A@, E C C (X, E). Consequently, when A is a function algebra, f (x) is a weak* -extreme point for all x in the Choquet boundary of A; the conclusion does not hold on the Silov boundary. ABSTRACT.

1. Introduction

For a Banach space E we denote by E 1 the closed unit ball in E and by 8eE1 the set of extreme points of E 1 . In 1961 Phelps [16] observed that for the space C(X) of all continuous functions on a compact Hausdorff space X every point fin Be (C (X)) 1 remains extreme when C (X) is canonically embedded into its second dual C (X)**. The question whether the same is true for any Banach space was answered in the negative by Y. Katznelson who showed that the disc algebra fails that property. A point x E 8eE1 is called weak* -extreme if it remains extreme in 8eEi*; we denote by a;E1 the set of all such points in E 1 . The importance of this class for geometry of Banach spaces was enunciated by Rosenthal when he proved that E has the Radon-Nikodym property if and only if under any renorming the unit ball of E has a weak*-extreme point [19]. While not all extreme points are weak* -extreme the later category is among the largest considered in the literature. For example we have: strongly exposed

0 precisely when T has closed range. Standard duality theory now leads to the following result. PROPOSITION 8. An operator T E L(X) is the restriction of an £(1')-scalar operator if and only if its adjoint T * is the quotient of an £ (1') -scalar operator. Moreover, ifT is the quotient of an £(1')-scalar operator, then T* is the restriction of an £(1')-scalar operator, and hence there exist constants c, s > 0 for which

for all n EN.

0

In general, it is not known if the last growth condition characterizes the quotients of £(1')-scalar operators, but, by Proposition 8 and [23, Prop.5], this is the case for the class of all unilateral weighted left shifts on £P(N 0 ) for arbitrary 1 < p < oo. More precisely, a unilateral weighted left shift on £P(N 0) satisfies the growth condition of Proposition 8 if and only if it admits a bilateral weighted shift lifting on £P(Z) that is £(1')-scalar. Similar results hold for more general growth conditions; see [22] and [23]. For instance, by another classical result due to Colojoara and Foia§, an invertible operator T E L(X) is decomposable provided that T satisfies Beurling's condition (B), in the sense that

[10, 5.3.2] and [18, 4.4.7]. Clearly, property (B) is inherited by restrictions, but it remains open, if every operator with property (B) has an invertible extension with property (B). In fact, it is not known, if property (B) implies property (jJ). For certain unilateral weighted right shifts, a positive answer was recently given in [22] and [23]. 3. Localization of the single-valued extension property For an arbitrary operator T E L(X) on a complex Banach space X, here the spaces K(T) := Xr(C\{0}) and H 0 (T) := Xr( {0}) will be of particular importance. Both spaces were, in some disguise, studied by Mbekhta and also by Vrbova; see [19], [20], and [30]. By [18, 3.3.7], K(T) coincides with the analytic core of T, defined to consist of all x E X for which there exist a constant c > 0 and elements Xn E X such that for all n EN. By this characterization and the open mapping theorem, K(T) =X if and only if T is surjective. In terms of local spectral theory, this follows also from the fact that Usu(T) is the union of all local spectra ofT. On the other hand, by [18, 3.3.13], H 0 (T) is the quasi-nilpotent part of T, defined as the set of all x E X for which

11Tnxll 1 /n __, 0

as n __,

oo.

In general, neither K(T) nor H 0 (T) need to be closed, but, if 0 is isolated in u(T), then, by [19, 1.6], both spaces are closed and X = K(T) EB H 0 (T). For more on operators with closed K(T) and H 0 (T), see [1], [2], and [24]. For instance, by [24, Cor.6], for any non-invertible decomposable operator T, the point 0 is isolated in


T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN

254

cr(T) precisely when K(T) is closed. In particular, the analytic core of a compact or, more generally, a Riesz operator Tis closed exactly when T has finite spectrum, [24, Cor.9]. The spaces H 0 (T) and K(T) are related to the kernel N(T) and the range R(T) of T as follows. With the notation

UN(Tn)

n 00

00

N(T) :=

and

R(T) :=

n=l

R(Tn)

n=l

for the generalized kernel and range ofT, there is an increasing chain of kernel-type spaces

N(T)

~

N(Tn)

~

N(T)

~

Ho(T)

~

Xr( {0})

and a decreasing chain of range-type spaces

R(T) ;:2 R(Tn) ;:2 R(T) ;:2 K(T) ;:2 Xr(0) for arbitrary n E N, [18, 1.2.16 and 3.3.1]. The geometric position of the kerneltype spaces versus the range-type spaces turns out to be intimately related to a certain localized version of SVEP for the operator T and its adjoint T*. An operator T E L(X) is said to have SVEP at a point >. E C, if, for every open disc U centered at >., the operator Tu is injective on H(U, X). This notion dates back to Finch [17], and was pursued further, for instance, in [1], [2], [3], [4], [5], and [20]. Evidently, T has SVEP at >. precisely when T - >. has SVEP at 0, while SVEP for T is equivalent to SVEP for T at >. for each >. E C. Local spectral theory leads to a variety of characterizations of this localized version of SVEP that involve the kernel-type and range-type spaces introduced above. Our starting point is the following characterization from [3, 1.9]. The result shows, in particular, that every injective operator T E L(X) has SVEP at 0, and may be viewed as a local version of the classical fact that T has SVEP if and only if Xr(0) = {0}, [18, 1.2.16]. For completeness, we include a short new proof that uses nothing but local spectral theory. THEOREM

9. For every operator T

E

L(X), the following equivalences hold:

T has SVEP at 0 ¢::> N(T) n Xr(0) = {0}

¢::>

crr(x) = {0} for all 0-=/- x E N(T).

Proof. First suppose that T has SVEP at 0, and consider an arbitrary x E N(T) for which crr(x) is empty. Then 0 E pr(x) so that there exists an f E H(U, X) on some open disc U with center 0 for which (T-.X)f(.X) = x for all>. E U. It follows that (T- >.)Tf(>.) = Tx = 0 for all>. E U, and therefore Tf(>.) = 0 for all>. E U, since T has SVEP at 0. Thus x = Tf(O) = 0, and hence N(T) n Xr(0) = {0}. Next observe that, for each x E N(T), the definition f(.X) := -xj>. yields an analytic function for which (T - .X)f(.X) = x for all non-zero >. E C. Thus crr(x) ~ {0} for all x E N(T). Consequently, the second and third assertions are equivalent. Finally suppose that N(T) n Xr(0) = {0}, let U be an open disc with center 0, and consider a function f E H(U, X) for which Tu f = 0. By [18, 1.2.14], err(!(>.))= crr(O) = 0 for all>. E U. Now, for the power series representation f(.X) = L:;:;:"= 0 an >. n for all >. E U, our task is to show that each of the coefficients an E X is zero. For the case n = 0, this is immediate, since a 0 = f(O) E N(T) nXr(0) = {0}. But then it follows that for all>. E U,


THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES

255

and therefore (T- A) (a 1 + a 2 A + a 3 A2 + · · · ) = 0 first for all non-zero A E U, and then, by continuity, also for A = 0. Exactly as before, we conclude that a 1 = 0 and hence, by induction, an = 0 for all n 2': 0. Thus f = 0 on U, as desired. 0 Since N(T) n K(T) ~ Xr( {0}) n Xr(C \ {0}) = Xr(0), it clearly follows that N(T) n K(T) = N(T) n Xr(0) for every T E L(X). Thus, by Theorem 9, T has SVEP at 0 if and only if N(T) n K(T) = {0}. In particular, if Tis surjective, then, as noted above, K(T) =X, so that T has SVEP at 0 precisely when Tis injective. This characterization from [3, 1.11] extends a classical result due to Finch [17]. As another immediate consequence of Theorem 9, we obtain the following result. COROLLARY 10. An operator T E L(X) has SVEP at 0 provided that either H 0 (T) n K(T) = {0} or N(T) n R(T) = (0). 0 Recent counter-examples in [2] show that, in general, none of the latter conditions is equivalent to SVEP ofT at 0, thus disproving a claim made in [20, 1.4]. However, by [1, 2.7], [5, 1.3], and Theorem 12 below, equivalences do hold for certain classes of operators. We now describe how the localized SVEP behaves under duality. For a linear subspace M of X, let MJ._ := {cp EX*: cp(x) = 0 for all x EM}, and for a linear subspace N of X*, let J._N := {x EX: cp(x) = 0 for all cp EN}. By the bipolar theorem, J._ ( M J._) is the norm-closure of M, and ( J._ N) J._ is the weak-* -closure of N. Moreover, for every T E L(X), it is well known that N(T*) = R(T)J._ and N(T) = J._R(T*), while R(T) is a norm-dense subspace of J._N(T*), and R(T*) is a weak-*-dense subspace of N(T)J._. An elementary short proof of the following result may be found in [2, 4.1].

(a) (b) (c)

PROPOSITION 11. For every operator T E L(X), the following assertions hold: K(T) ~ J._H0 (T*) and K(T*) ~ H 0 (T)J._; if H 0 (T) + R(T) is norm-dense in X, then T* has SVEP at 0; if H 0 (T*) + R(T*) is weak-*-dense in X*, then T has SVEP at 0. 0

Even in the Hilbert space setting, the inclusions in part (a) of Proposition 11 need not be identities, and the implications of parts (b) and (c) cannot be reversed in general; see [2] for counter-examples in the class of weighted shifts. However, for suitable classes of operators, the results can be improved. As usual, an operator T E L(X) is said to be a semi-Fredholm operator, if either N(T) is finite-dimensional and R(T) is closed, or R(T) is of finite codimension in X. Also, an operator T E L(X) is said to be semi-regular, if R(T) is closed and N(T) ~ R(T); see [18], [19], and [21] for a discussion of these operators. THEOREM 12. Suppose that the operator T E L(X) is either semi-Fredholm or semi-regular. Then the following assertions hold: (a) R(T) = K(T) = J._Ho(T*) = J._N(T*); (b) R(T*) = K(T*) = H 0 (T)J._ = N(T)J._; (c) N(T) n R(T) = {0} ¢? T has SVEP at 0; (d) N(T*) n R(T*) = {0} ¢;> T* has SVEP at 0; (e) N(T) + R(T) =X¢? Ho(T) + R(T) =X ¢;> T* has SVEP at 0; (f) N(T*) + R(T*) =X* ¢? H 0 (T*) + R(T*) =X* ¢;> T has SVEP at 0, where w* indicates the closure with respect to the weak- *-topology. 0 77~~~~~w*

w*


256

T.

L. MILLER,

V. G.

MILLER, AND M. M. NEUMANN

Theorem 12 was recently obtained in [2], see also [5]. An important ingredient of the proof is the fact that T has SVEP at 0 if and only yn has SVEP at 0 for arbitrary n E N. This equivalence is a special case of a spectral mapping formula for the set 6(T) of all.\ E Cat which T fails to have SVEP, namely 6(J(T)) = f(6(T)) for every analytic function f on some open neighborhood of CJ(T); see [2, 3.1] and also Theorem 18 below. Further developments may be found in [1], [3], [4], [5], [17], and [20]. Here we mention only one simple consequence of Theorem 12 for semi-regular operators from [3, 2.13]. ForTE L(X), let PK(T) consist of all A E C for which T- A is semi-regular. The Kato spectrum CJK(T) := C \ PK(T) is a closed subset of CJ(T) and contains OCJ(T); see [18, 3.1] and [21] for details. We include a short proof of the following result, since the dichotomy for the connected components of the Kato resolvent set PK(T) with respect to the localized SVEP will play an essential role in Section 4. THEOREM 13. LetT E L(X) be semi-regular. Then T has SVEP at 0 precisely when T is injective, or, equivalently, when T is bounded below, while T* has SVEP at 0 precisely when T is surjective. Moreover, for arbitrary T E L(X), each connected component n of PK(T) satisfies either n ~ 6(T) or n n 6(T) = 0. The inclusion rl ~ 6(T) occurs precisely when n ~ CJp(T), or, equivalently, when n n CJap(T) -=/= 0, while the identity rl n 6(T) = 0 occurs precisely when rl n CJp(T) = 0, or, equivalently, when n \ CJap(T)-=/= 0.

Proof. If T is semi-regular, then N(T) n R(T) = N(T) and N(T) + R(T) = R(T) = R(T). Hence the first assertions follow from parts (c) and (e) of Theorem 12. For the last claim, it suffices to see that injectivity ofT-.\ for some.\ En entails that T- J-L is injective for every J-L E fl. But this is clear, since, by part (b) of Theorem 12, N(T- J-L) = j_R(T*- J-L) and, by [18, 3.1.6 and 3.1.11], R(T*- J-L) = R(T*- .\) for all J-L E rl. D It is well known that the approximate point spectrum and the surjectivity spectrum of an arbitrary operator T E L(X) are related by the duality formulas CJap(T) = CJ8 u(T*) and CJ8 u(T) = CJap(T*), [18, 1.3.1]. Moreover, by [18, 1.3.2 and 3.1.7], CJ8 u(T) = CJ(T) and CJap(T) = CJK(T) if T has SVEP, and CJap(T) = CJ(T) and CJsu(T) = CJK(T) if T* has SVEP. The following local version of these results is immediate from Theorem 13. PROPOSITION 14. For every operator T E L(X), the following assertions hold: (a) If.\ E CJ(T) \ CJap(T), then T has SVEP at.\, butT* fails to have SVEP at.\; (b) if.\ E CJ(T) \ CJsu(T), then T* has SVEP at.\, butT fails to have SVEP at.\. D

The next result from [2, 5.2] is a straightforward consequence of Proposition 14. For instance, it follows that 6(T*) is the open unit disc for every non-invertible operator T with property (P) or (B). Further examples including analytic Toeplitz operators, composition operators on Hardy spaces, and weighted shifts may be found in [2]. CoROLLARY 15. If CJap(T) ~ OCJ(T), then T has SVEP and 6(T*) = int CJ(T). Similarly, if CJ8 u(T) ~ OCJ(T), then T* has SVEP and 6(T) = intCJ(T). D


THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES

257

4. Localization of the properties ((3) and (J) There is a natural extension of the class of decomposable operators for which spectral decompositions are only required with respect to a given open subset U of the complex plane. These operators were introduced by Vasilescu as residually decomposable operators in 1969, shortly after the publication of the seminal monograph [10]. They became also known asS-decomposable operators with S = C \ U, and were studied by Bacalu, Nagy, Vasilescu, and others; see [29, Ch.4]. As in [6] and [13], we now say that an operator T E L(X) on a complex Banach space X is decomposable on an open subset U of C provided that, for every finite open cover {V1, ... , Vn} of C with C \ U ~ V1, there exist X 1, ... , Xn E Lat(T) with the property that X=

xl + ... + Xn

and

O'(T I Xk)

~ vk

fork= 1, ... 'n.

It is known, although certainly not obvious, that, in this definition, it suffices to consider the case n = 2; see [6] and [29]. Evidently, classical decomposability occurs when U =C. On the other hand, every operator T E L(X) is at least decomposable on its resolvent set p(T). Among the remarkable early accomplishments of the theory is the following result due to Nagy [25] from 1979: For every T E L(X), there exists a largest open set U ~ Con which Tis decomposable. The complement of this set is Nagy's spectral residuum Sr(T), a closed, possibly empty, subset of O'(T). In the present section, we shall employ the recent results of Albrecht and Eschmeier [6] to obtain a short proof for the existence and a useful description of Nagy's spectral residuum. In particular, we shall see how Sr(T) is related to the Kato spectrum O'K(T) and the essential spectrum O'e(T). For this, we shall work with certain localized versions of property ((3) and property (J) from [6]. An operator T E L(X) is said to possess Bishop's property ((3) on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective with closed range, equivalently, if, for every sequence of analytic functions f n : V -+ X for which (T- >..) f n ( >..) -+ 0 as n -+ oo locally uniformly on V, it follows that fn(>..) -+ 0 as n -+ oo, again locally uniformly on V. It is straightforward to check that this condition is preserved under arbitrary unions of open sets. This shows that there exists a largest open set on which T has property ((3), denoted by U(3(T). Its complement Sf3(T) := C \ Uf3(T) is a closed, possibly empty, subset of O"(T). In fact, T satisfies Bishop's classical property ((3) precisely when S(3(T) = 0. Moreover, the operator T is said to have property ( J) on U, if X

= Xr(C \ V) + Xr(W)

for all open sets V, W ~ C for which C \ U ~ V ~ V ~ W; see [6] and [13]. Quite remarkably, as shown in [6, Th.3], this condition holds precisely when, for each closed set F ~ C and every finite open cover {V1 , ... , Vn} ofF with F \ U ~ V1 , it follows that Xr(F) ~ Xr(VI) + · · · + Xr(V n); see also [18, 2.2.2] for the case

U=C.

These localized versions of ((3) and (J) already proved to be useful in the theory of invariant subspaces for operators on Banach spaces, [14]. The following result summarizes the main accomplishments from [6].


258

T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN

THEOREM 16. For every operator T following equivalences hold:

(a) (b) (c) (d) (e)

T has (/3) on U

¢:?

T* has (8) on U;

T has (8) on U

¢:?

T* has (/3) on U;

T is decomposable on U T has (/3) on U T has (8) on U

¢:? ¢:?

¢:?

E

L(X) and every open set U

C such that (1,m) =[[mil= 1. For any function f on G and for any x E G, we write Lxf for the left translate of f by x, i.e. (Lxf)(y) := f(xy) for y E G. DEFINITION 1.1. A locally compact group G is called amenable if there is a (left) translation invariant mean on G, i.e. a mean m such that

(¢, m) = (Lx¢, m)

(¢

E

L 00 (G), x E G).

1.2. (1) Since the Haar measure of a compact group G is finite, L 00 (G) C L 1 (G) holds. Consequently, Haar measure is an invariant mean on G. (2) For abelian G, the Markov-Kakutani fixed point theorem yields an invariant mean on G. (3) The free group in two generators is not amenable ([Pat, (0.6) Example]).

EXAMPLE

Moreover, amenability is stable under standard constructions on locally compact groups such as taking subgroups, quotients, extensions, and inductive limits. Amenable, locally compact groups were first considered by J. v. Neumann ([Neu]) in the discrete case; he used the term "Gruppen von endlichem MaB". The adjective amenable for groups satisfying Definition 1.1 is due to M. M. Day ([Day]), apparently with a pun in mind: They are amenable because they have an invariant mean, but also since they are particularly pleasant to deal with and thus are truly amenable - just in the sense of that adjective in everyday speech. For more on the theory of amenable, locally compact groups, we refer to the monographs [Gre], [Pat], and [Pie]. 2. homological algebra, ...

We will not attempt here to give a survey on a area as vast as homological algebra, but outline only a few, basic cohomological concepts that are relevant in connection with abstract harmonic analysis. For the general theory of homological algebra, we refer to [C-E], [MacL], and [Wei]. The first to adapt notions from homological algebra to the functional analytic context was H. Kamowitz in [Kam]. Let 2t be a Banach algebra. A Banach 2t-bimodule is a Banach space E which is also an 2t-bimodule such that the module actions of 2t on E are jointly continuous.


ABSTRACT HARMONIC ANALYSIS

265

A derivation from Qt to E is a (bounded) linear map D: Qt----+ E satisfying

D(ab) =a· Db+ (Da) · b

(a, bE Qt);

the space of all derivation from Qt to E is commonly denoted by Z 1 (Qt, E). A derivation D is called inner if there is x E E such that

Da =a· x- x ·a

(a E Qt).

The symbol for the subspace of Z 1 (Qt, E) consisting of the inner derivations is 8 1 (Qt, E); note that 8 1 (Qt, E) need not be closed in Z 1 (Qt, E). DEFINITION 2.1. Let Qt be a Banach algebra, and let E be a Banach Qt-bimodule. Then then the first Hochschild cohomology group 7i 1 (Qt, E) of Qt with coefficients in E is defined as

7i 1 (Qt, E):= Z 1 (Qt, E)/B 1 (Qt, E). The name Hochschild cohomology group is in the honor of G. Hochschild who first considered these groups in a purely algebraic context ([Hoch 1] and

[Hoch 2]).

Given a Banach Qt-bimodule E, its dual space E* carries a natural Banach Qt-bimodule structure via

(x, a·¢) := (x ·a,¢)

and

(x,

(1r(x)~,

77)

E 5) is called a coefficient function of 1r.

ExAMPLE 2.11. The left regular representation >. of G on L 2 (G) is given by .A(x)~

:= Lx-1~

(x E G, ~ E L 2 (G)).

DEFINITION 2.12 ([Eym]). Let G be a locally compact group. (a) The Fourier algebra A( G) of G is defined as

A (G) := {!: G

----+

C : f is a coefficient function of >.}.

(b) The Fourier-Stieltjes algebra B( G) of G is defined as B( G) := {f: G----+ C : f is a coefficient function of a unitary representation of G}. It is immediate that A( G) C B(G), that B(G) consists of bounded continuous functions, and that A( G) C C0 (G). However, it is not obvious that A( G) and B(G) are linear spaces, let alone algebras. Nevertheless, the following are true ([Eym]): • Let C* (G) be the enveloping C* -algebra of the Banach *-algebra L 1 (G). Then B(G) can be canonically identified with C*(G)*. This turns B(G) into a commutative Banach algebra. • Let VN(G) :=>.(G)" denote the group von Neumann algebra of G. Then A( G) can be canonically identified with the unique predual of VN( G). This turns A( G) into a commutative Banach algebra whose character space is G. • A(G) is a closed ideal in B(G). If G is an abelian group with dual group r, then the Fourier and FourierStieltjes transform, respectively, yield isometric isomorphisms A( G) ~ L 1 (f) and B(G) ~ M(f). Consequently, A(G) is amenable for any abelian locally compact group G. It doesn't require much extra effort to see that A( G) is also amenable if G has an abelian subgroup of finite index ([L-L-W, Theorem 4.1] and [For 2, Theorem 2.2]). On the other hand, every amenable Banach algebra has a bounded approximate identity, and hence Leptin's theorem ([Lep]) implies that the amenability of A( G) forces G to be amenable. Nevertheless, the tempting conjecture that A( G) is amenable if and only if G is amenable is false:


VOLKER RUNDE

268

THEOREM 2.13 ([Joh 4]). The Fourier algebra of 80(3) is not amenable. This leaves the following intriguing open question: QUESTION 2.14. Which are the locally compact groups G for which A(G) is amenable? The only groups G for which A( G) is known to be amenable are those with an abelian subgroup of finite index. It is a plausible conjecture that these are indeed the only ones. The corresponding question for weak amenability is open as well. B. E. Forrest has shown that A( G) is weakly amenable whenever the principal component of G is abelian ([For 2, Theorem 2.4]). One can, of course, ask the same question(s) for the Fourier-Stieltjes algebra: QuESTION 2.15. Which are the locally compact groups G for which B(G) is amenable? Here, the natural conjecture is that those groups are precisely those with a compact, abelian subgroup of finite index. Since A( G) is a complemented ideal in B(G), the hereditary properties of amenability for Banach algebras ([Run, Theorem 2.3.7]) yield that A(G) has to be amenable whenever B(G) is. It is easy to see that, if the conjectured answer to Question 2.14 is true, then so is the one to Question 2.15. Partial answers to both Question 2.14 and Question 2.15 can be found in [L-L-W] and [For 2].

3. and operator spaces Given any linear space E and n EN, we denote then x n-matrices with entries from E by Mn(E); if E = C, we simply write Mn. Clearly, formal matrix multiplication turns Mn(E) into an Mn-bimodule. Identifying Mn with the bounded linear operators on n-dimensional Hilbert space, we equip Mn with a norm, which we denote by I · I· DEFINITION 3.1. An operator space is a linear space E with a complete norm II · lin on Mn(E) for each n EN such that

I ~I~

(R 1)

lln+m = max{llxlln, IIYIIm}

(n,m EN,

X

E Mn(E), y E Mm(E))

and

(R 2)

(n EN, x E Mn(E), a, (3 E Mn)·

EXAMPLE 3.2. Let Sj be a Hilbert space. The unique C*-norms on Mn(B(SJ))

~

B(SJn) turn B(SJ) and any of its subspaces into operator spaces.

Given two linear spaces E and F, a linear map T: E -+ F, and n E N, we define the the n-th amplification T(n) : Mn(E) -+ Mn(F) by applying T to each matrix entry. DEFINITION 3.3. Let E and F be operator spaces, and letT E B(E, F). Then: (a) T is completely bounded if IITIIcb := sup IIT(n) lls(Mn(E),Mn(F)) < nEN

00.



269

(b) T is a complete contraction if IITIIcb :::; 1. (c) Tis a complete isometry if T(n) is an isometry for each n EN. The following theorem due to Z.-J. Ruan marks the beginning of abstract operator space theory: THEOREM 3.4 ([Rua 1]). Let E be an operator space. Then there is a Hilbert space ,fj and a complete isometry from E into B(SJ). To appreciate Theorem 3.4, one should think of it as the operator space analogue of the elementary fact that every Banach space is isometrically isomorphic to a closed subspace of C(O) for some compact Hausdorff space n. One could thus define a Banach space as a closed subspace of C(O) some compact Hausdorff space 0. With this definition, however, even checking, e.g., that £1 is a Banach space or that quotients and dual spaces of Banach spaces are again Banach spaces is difficult if not imposssible. Since any C* -algebra can be represented on a Hilbert space, each Banach space E can be isometrically embedded into B(SJ) for some Hilbert space ,fj. For an operator space, it is not important that, but how it sits inside B(SJ). There is one monograph devoted to the theory of operator spaces ([E-R]) as well as an online survey article ([Wit et al.]). The notions of complete boundedness as well as of complete contractivity can be defined for multilinear maps as well ([E-R, p. 126]). Since this is somewhat more technical than Definition 3.3, we won't give the details here. As in the category of Banach spaces, there is a universallinearizer for the right, i.e. completely bounded, bilinear maps: the projective operator space tensor product ([E-R, Section 7.1]), which we denote by~. DEFINITION 3.5. An operator space Q( which is also an algebra is called a completely contractive Banach algebra if multiplication on Q( is a complete (bilinear) contraction. The universal property of~ ([E-R, Proposition 7.1.2]) yields that, for a completely contractive Banach algebra Q(, the multiplication induces a complete (linear) contraction Ll: Q(~Q( ---+ Qt. EXAMPLE 3.6. (1) For any Banach space E, there is an operator space maxE such that, for any other operator space F, every T E B(E, F) is completely bounded with IITIIcb = IITII ([E-R, pp. 47-54]). Given a Banach algebra Q(, the operator space max Q( is a completely contractive Banach algebra ([E-R, p. 316]). (2) Any closed subalgebra of B(SJ) for some Hilbert space ,fj is a completely contractive Banach algebra. To obtain more, more interesting, and ~ in the context of abstract harmonic analysis ~ more relevant examples, we require some more operator space theory. Given two operator spaces E and F, let

CB(E, F)

:=

{T: E---+ F : Tis completely bounded}.

It is easy to check that CB(E, F) equipped with ll·llcb is a Banach space. To define an operator space structure on CB(E, F), first note that Mn(F) is, for each n EN, an operator space in a canonical manner. The (purely algebraic) identification

Mn(CB(E, F)):= CB(E, Mn(F))

(n EN)


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then yields norms ll·lln on the spaces Mn(CB(E,F)) that satisfy (R 1) and (R 2), which is not hard to verify. Since, for any operator space E, the Banach spaces E* and CB(E, C) are isometrically isomorphic ([E-R, Corollary 2.2.3]), this yields a canonical operator space structure on the dual Banach space of an operator space. In partiuclar, the unique predual of a von Neumann algebra is an operator space in a canonical way. We shall see how this yields further examples of completely contractive Banach algebras. We denote the W* -tensor product by ®. DEFINITION 3.7. A Hopf-von Neumann algebra is a pair (9J1, \7), where 9J1 is a von Neumann algebra, and \7 is a co-multiplication: a unital, injective, w*continuous *-homomorphism \7: 9J1 __, 9J1@9J1 which is co-associative, i.e. the diagram

9J1

V' --------*

9J1®9J1

1

V'@id!JR

V'l

9J1@9J1

id!JR @V'

9J1@9J1@9J1

commutes. EXAMPLE 3.8. Let G be a locally compact group. (1) Define \7: L 00 (G) __, L 00 (G x G) by letting

(\7¢)(xy) := ¢(xy)

(¢

E L 00 (G),

x, y E G).

Since L 00 (G)®L 00 (G) ~ L 00 (G x G), this turns L 00 (G) into a Hopf-von Neumann algebra. (2) Let W*(G) be the enveloping von Neumann algebra of C*(G). There is a canonical w* -continuous homomorphism w from G into the unit aries of W* (G) with the following universal property: For any unitary representation 7r of G on a Hilbert space, there is unique w* -continuous *homomorphism e : W* (G) __, 7f (G) II such that 7f = e 0 w. Applying this universal property to the representation G--> W*(G)®W*(G),

x

f---7

w(x)@ w(x)

yields a co-multi plication \7 : W* (G) __, W* (G)® W* (G). Given two von Neumann algebras 9J1 and 1)1 with preduals 9J1. and !Jt., their W* -tensor product 9J1@1)1 also has a unique predual (9J1®1J1).. Operator space theory allows to identify this predual in terms of 9J1. and !Jt. ([E-R, Theorem 7.2.4]): Since VN( G)® VN(H) ~ implies in particular that

(9J1®1J1). ~ 9J1.®1J1 •. VN( G x H) for any locally compact groups G and H, this A(G x H)~

A(G)@A(H).

Suppose now that 9J1 is a Hopf-von Neumann algebra with predual 9J1 •. The comultiplication \7 : 9J1 __, 9J1®9J1 is w* -continuous and thus the adjoint map of a complete contraction \7 * : 9J1. ®9J1. __, 9J1.. This turns 9J1. into a completely contractive Banach algebra. In view of Example 3.8, we have:



271

EXAMPLE 3.9. Let G be a locally compact group. (1) The multiplication on L 1 (G) induced by \7 as in Example 3.8.1 is just the usual convolution product. Hence, L 1 (G) is a completely contractive Banach algebra. (2) The multiplication on B( G) induced by \7 as in Example 3.8.2 is pointwise multiplication, so that B( G) is a completely contractive Banach algebra. Since A( G) is an ideal in B( G) and since the operator space strucures A( G) has as the predual of VN(G) and as a subspce of B(G) coincide, A(G) with its canonical operator space structure is also a completely contractive Banach algebra. REMARK 3.10. Since A( G) fails to be Arens regular for any non-discrete or infinite, amenable, locally compact group G ([For 1]), it cannot be a subalgebra of the Arens regular Banach algebra B(fJ). Hence, for those groups, A( G) is not of the form described in Example 3.6.2. We now return to homological algebra and its applications to abstract harmonic analysis. An operator bimodule over a completely contractive Banach algebra m is an operator space E which is also an m-bimodule such that the module actions of m on E are completely bounded bilinear maps. One can then define operator Hochschild cohomology groups OH 1 (m, E) by considering only completely bounded derivations (all inner derivations are automatically completely bounded). It is routine to check that the dual space of an operator m-bimodule is again an operator m-bimodule, so that the following definition makes sense: DEFINITION 3.11 ([Rua 2]). A completely contractive Banach algebra m is called operator amenable if OH 1 (m, E*) = {0} for each operator m-bimodule E. The following result ([Rua 2, Theorem 3.6]) shows that Definition 3.11 is indeed a good one: THEOREM 3.12 (Z.-J. Ruan). Let G be a locally compact group. amenable if and only if A( G) is operator amenable.

Then G is

REMARK 3.13. A Banach algebra m is amenable if and only if max m is operator amenable ([E-R, Proposition 16.1.5]). Since L 1 (G) is the predual of the abelian von Neumann algebra L 00 (G), the canonical operator space structure on L 1 (G) is maxL 1 (G). Hence, Definition 3.11 yields no information on L 1 (G) beyond Theorem 2.2. The following is an open problem: QuESTION 3.14. Which are the locally compact groups G for which B(G) is operator amenable? With Theorem 2.8 and the abelian case in mind, it is reasonable to conjecture that B( G) is operator amenable if and only if G is compact. One direction is obvious in the light of Theorem 3.12; a partial result towards the converse is given in [R-Sp]. Adding operator space overtones to Definition 2.5, we define: DEFINITION 3.15. A completely contractive Banach algebra miscalled operator weakly amenable if OH 1 (m, m*) = {0}.


VOLKER RUNDE

272

In analogy with Theorem 2. 7, we have: THEOREM 3.16 ([Spr]). Let G be a locally compact group. Then A(G) is operator weakly amenable.

One can translate Helemski1's homological algebra for Banach algebras relatively painlessly to the operator space setting: This is done to some extent in [Ari] and [Woo 1]. Of course, appropriate notions of projectivity and flatness play an important role in this operator space homological algebra. Operator biprojectivity and bifiatness can be defined as in the classical setting, and an analogue - with ® instead of ®I' - of the characterization used for Definition 2.9 holds. The operator counterpart of Theorem 2.10 was discovered, independently, by 0. Yu. Aristov and P. J. Wood: THEOREM 3.17 ([Ari], [Woo 2]). Let G be a locally compact group. Then G is discrete if and only if A( G) is operator biprojective.

As in the classical setting, both operator amenability and operator biprojectivity imply operator biftatness. Hence, Theorem 3.17 immediately supplies examples of locally compact groups G for which A( G) is operator biftat, but not operator amenable. A locally compact group is called a [SIN]-group if L 1 (G) has a bounded approximate identity belonging to its center. By [R-X, Corollary 4.5], A( G) is also operator bifiat whenever G is a [SIN]-group. It may be that A( G) is operator biftat for every locally compact group G: this question is investigated in [A-R-Sp]. All these results suggest that in order to get a proper understanding of the Fourier algebra and of how its cohomological properties relate to the underlying group, one has to take its canonical operator space structure into account.

References [Ari] [A-R-Sp] [B-C-D] [C-E] [C-H] [D-Gh-H] [Day] [D-Gh] [E-R] [Eym] [Fol] [For 1] [For 2] [Gre]

0. Yu. Aristov, Biprojective algebras and operator spaces. J. Math. Sci. (to appear). 0. Yu. Aristov, V. Runde, and N. Spronk. Operator biflatness of the Fourier algebra.

In preparation. W. G. Bade, P. C. Curtis, Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. (3) 55 (1987), 359-377. H. Cartan and S. Eilenberg, Homological algebro. Princeton University Press, Princeton, 1956. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96 (1989), 507-549. H. G. Dales, F. Ghahramani, and A. Ya. Helemski1, The amenability of measure algebras. J. London Math. Soc. 66 (2002), 213-226. M. M. Day, Means on semigroups and groups. Bull. Amer. Math. Soc. 55 (1949), 1054-1055. M. Des pic and F. Ghahramani, Weak amenability of group algebras of locally compact groups. Canad. Math. Bull. 37 (1994), 165-167. E. G. Effros and Z.-J. Ruan, Operotor spaces. Clarendon Press, Oxford, 2000. P. Eymard, L'algebre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181-236. G. B. Folland, A course in abstroct harmonic analysis. CRC Press, Boca Raton, Florida, 1995. B. E. Forrest, Arens regularity and discrete groups. Pacific J. Math. 151 (1991), 217-227. B. E. Forrest, Amenability and weak amenability of the Fourier algebra. Preprint (2000). F. P. Greenleaf, Invariant means on locally compact groups. Van Nostrand, New York-Toronto-London, 1969.



[Gui]

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A. Guichardet, Sur l'homologie et Ia cohomologie des algebres de Banach. C. R. Acad. Sci. Paris, Ser. A 262 (1966), 38-42. C. S. Herz, Harmonic synthesis for subgruops. Ann. Inst. Fourier (Grenoble) 23 [Her] (1973)' 91-123. [H-R] E. Hewitt and K. A. Ross, Abstract harmonic analysis, I and II. Springer Verlag, Berlin-Heideberg-New York, 1963 and 1970. [Hell] A. Ya. Helemskil, Flat Banach modules and amenable algebras. Trans. Moscow Math. Soc. 47 (1985), 199-224. [Hel2] A. Ya. Helemskil, The homology of banach and topological algebras (translated from the Russian). Kluwer Academic Publishers, Dordrecht, 1989. [Hoch 1] G. Hochschild, On the cohomology groups of an associative algebra. Ann. of Math. (2) 46 (1945), 58-67. [Hoch 2] G. Hochschild, On the cohomology theory for associative algebras. Ann. of Math. (2) 47 (1946), 568-579. [Joh 1] B. E. Johnson, Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). B. E. Johnson, Derivations from L 1 (G) into L 1 (G) and L 00 (G). In: J.P. Pier (ed.), [Joh 2] Harmonic analysis (Luxembourg, 1987}, pp. 191-198. Lectures Notes in Mathematics 1359. Springer Verlag, Berlin-Heidelberg-New York, 1988. [Joh 3] B. E. Johnson, Weak amenability of group algebras. Bull. London Math. Soc. 23 (1991), 281-284. B. E. JOHNSON, Non-amenability of the Fourier algebra of a compact group. J. London [Joh 4] Math. Soc. (2) 50 (1994), 361-374. H. Kamowitz, Cohomology groups of commutative Banach algebras. Trans. Amer. [Kam] Math. Soc. 102 (1962), 352-372. [L-L-W] A. T.-M. Lau, R. J. Loy, and G. A. Willis, Amenability of Banach and C*-algebras on locally compact groups. Studia Math. 119 (1996), 161-178. [Lep] H. Leptin, Sur l'algebre de Fourier d'un groupe localement compact. C. R. Acad. Sci. Paris, Ser. A 266 (1968), 1180-1182. S. MacLane, Homology. Springer Verlag, Berlin-Heidelberg-New York, 1995. [MacL] J. von Neumann, Zur allgemeinen Theorie des MaBes. Fund. Math. 13 (1929), 73-116. [Neu] A. L. T. Paterson, Amenability. American Mathematical Society, Providence, 1988. [Pat] J. P. Pier, Amenable locally compact groups. Wiley-Interscience, New York, 1984. [Pie] [R-St] H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups. Clarendon Press, Oxford, 2000. [Rua 1] Z.-J. Ruan, Subspaces of C*-algebras. J. Funct. Anal. 76 (1988), 217-230. Z.-J. Ruan, The operator amenability of A(G). Amer. J. Math. 117 (1995), 1449[Rua 2] 1474. [R-X] Z.-J. Ruan and G. Xu, Splitting properties of operator bimodules and operator amenability of Kac algebras. In: A. Gheondea, R. N. Gologan and D. Timotin, Operator theory, operator algebras, and related topics, pp. 193-216. The Theta Foundation, Bucharest, 1997. V. Runde, Lectures on amenability. Lecture Notes in Mathematics 1774. Springer [Run] Verlag, Berlin-Heidelberg-New York, 2002. V. Runde and N. Spronk, Operator amenability of Fourier-Stieltjes algebras. Preprint [R-Sp] (2001). N. Spronk, Operator weak amenability of the Fourier algebra. Proc. Amer. Math. [Spr] Soc. 130 (2002), 3609-3617. [Tay] J. A. Taylor, Homology and cohomology for topological algebras. Adv. in Math. 9 (1970)' 137-182. C. A. Weibel, An introduction to homological algebra. Cambridge University Press, [Wei] Cambridge, 1994. [Wit et al.] G. Wittstock et al., What are operator spaces? - An online dictionary. URL: http://www.math.uni-sb.de/~ag-wittstock/projekt2001.html (2001). P. J. Wood, Homological algebra in operator spaces with applications to harmonic [Woo 1] analysis. Ph.D. thesis, University of Waterloo, 1999. P. J. Wood, The operator biprojectivity of the Fourier algebra. Canadian J. Math. [Woo 2] (to appear).


274

VOLKER RUNDE DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES, UNIVERSITY OF ALBERTA, ED-

T6G 2Gl E-mail address: vrunde!Dualberta. ca

MONTON, AB, CANADA



Relative Tensor Products for Modules over von Neumann Algebras David Sherman We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as composition of unbounded operators, noting that these ideas work equally well for LP modules. Finally, we prove some new results characterizing preclosedness of the map(~, 1J) >-> ~ @cp 1J· ABSTRACT.

1. Introduction

The purpose of this article is to summarize and explore some of the various constructions of the relative tensor product (RTP) of von Neumann algebra modules. Alternately known as composition or fusion, RTPs are a key tool in subfactor theory and the study of Morita equivalence. The idea is this: given a von Neumann algebra M, we want a map which associates a vector space to certain pairs of a right M-module and a left M-module. If we write module actions with subscripts, we have

(XM, M!V)

f---7

X ®M !V.

This should be functorial, covariant in both variables, and appropriately normalized. Other than this, we only need to specify which modules and spaces we are considering. In spirit, RTPs are algebraic; a ring-theoretic definition can be found in most algebra textbooks. But in the context of operator algebras, the requirement that the output be a certain type of space - typically a Hilbert space - causes an analytic obstruction. As a consequence, there are domain issues in any vector-based construction. Fortunately, von Neumann algebras have a sufficiently simple representation theory to allow a recasting of RTPs in algebraic terms. The analytic study of RTPs can be related nicely to noncommutative LP spaces. Indeed, examination of the usual (L 2 ) case reveals that the technical difficulties 2000 Mathematics Subject Classification. Primary: 46Ll0; Secondary: 46M05. Key words and phrases. relative tensor product, von Neumann algebra, bimodule.

275

©



276

DAVID SHERMAN

come from a "change of density". (We say that the density of an £P-type space is 1/p.) Once this is understood, it is easy to handle LP modules [JS] as well. Modular algebras ([Y], [S]) provide an elegant framework, so we briefly explain their meaning. The final section of the paper investigates the question, "When is the map (~, ry) f---> ~ ®'P 7J preclosed?" This may be considered as an extension of Falcone's theorem [F], in which he found conditions for the map to be everywhere-defined. We consider a variety of formulations. We have tried to make the paper as accessible as possible to non-operator algebraists, especially in the first half. Of course, even at this level many results rely on familiarity with the projection theory of von Neumann algebras; basic sources are [Tl], [T2], [KR]. Primary references for RTPs are [Sa], [P], [F], [C2].

2. Notations and background The basic objects of this paper are von Neumann algebras, always denoted here by M, N, or P. These can be defined in many equivalent ways: • C*-algebras which are dual spaces. • strongly-closed unital *-subalgebras of B(f:J). B(f:J) is the set of bounded linear operators on a Hilbert space f); the strong topology is generated by the seminorms x f---> I x~ II, ~ E f); the * operation is given by the operator adjoint. • *-closed subsets of B(f:J) which equal their double (iterated) commutant. The commutant of a setS C B(f:J) is {x E B(fJ) I xy = yx, "iy E S}. As one might guess from the definitions, the study of von Neumann algebras turns on the interplay between algebraic and analytic techniques. Finite-dimensional von Neumann algebras are direct sums of full matrix algebras. At the other extreme, commutative von Neumann algebras are all of the form L 00 (X, J.L) for some measure space (X, J.L), so the study of general von Neumann algebras is considered "noncommutative measure theory." Based on this analogy, the (unique) predual M* of Miscalled L 1 (M); it is the set of normal(= continuous in yet another topology, the a-weak) linear functionals on M c B(f:J), and can be thought of as "noncommutative countably additive measures". A functional
0::::} .)} be as before. Now by assumption, the function

z(w) = max{1/n I T(w)(e(1/n)(w)) < 1} is a.e. defined, non-zero, and finite. It is measurable by construction, so z and z- 1 represent elements of the extended center. Now write 'P = (Tz)z-lh· Let f be the


DAVID SHERMAN

290

spectral projection of z- 1h for [0,1]. We have f(w) = e(z(w))(w), so T(w)(f(w)) < 1. Then iJ>(f) < 1, and Tz(f) = i(ziJ>(f)) < 7'(1) < oo. Since f was a spectral projection of z- 1 h, we conclude that (z- 1 h)- 1 is Tz-measurable. 0 PROPOSITION

f:JM. The map

7.4. Let M be a factor with left module

M.!{

and right module

(7.8) is preclosed only under the same conditions as in Theorem 7.1; i.e. M semifinite and h- 1 is T-measurable, where
0 there is an L > 0 such that within every interval I C ~' III 2 L there is an x E I such that max lf(t)- f(t + x)l < f (H. Bohr [4]). According to the tEIR

famous theorem of Bochner [3], f is almost periodic on ~ if and only if the set of all the its translates ft(x) = f(x + t), t E ~is relatively uniformly compact in BC(~), space of bounded continuous functions on R Equivalently, f is almost periodic if it can be approximated uniformly on ~ by exponential polynomials, i.e. by functions n

of type L

akeiskx, where ak are complex, and sk are real numbers. It is easy to

k=l

see that the set AP(~) of all almost periodic functions on ~ is an algebra over C. Actually, under the uniform norm AP(~) is a commutative Banach algebra with unit.

.

Dirichlet coefficients a{, ..\ E ~of an almost periodic function f(x) on~ are 1 the numbers a{= lim f(x)e-,>-.xdx, where the limit, and its value in the

ly+T

T y right hand side exists independently on y E R Dirichlet coefficients a{ are nonzero for countably many ..\'sat most, which are called Dirichlet exponents of f(x). The set sp (f) of all Dirichlet's exponents of f(x) is called the spectrum of f. Hence, sp (f) = {..\ E ~ : a{ -=/= 0} is a countable set. It is customary to express the fact that ..\k are the Dirichlet exponents of f(x) and the numbers Ak = are the Dirichlet coefficients of f(x) for any k = 1, 2, ... by a power series notation, namely T-+oo

aC

00

f(x) "'LAkei>-.kx. This series, not necessarily convergent, is called the Dirichlet k=l

series of f(x). If all Dirichlet coefficients of a f E AP(~)

are zero, then, as it is easy to see, f = 0. Consequently, the correspondence between almost periodic functions and their Dirichlet series is injective. E~ery almost periodic function f on ~ can be extended as a continuous function f on the Bohr compactification ;3~ of R The Fourier coefficients of the extended in this way function on ;3~ equal the Dirichlet coefficients Ak of f. Moreover, the maximal ideal space MAP(JR) of the algebra of almost periodic functions on ~ is homeomorphic to the Bohr compactification ;3~ of R

J

c{

For every A c

~'

by

APA(~)

we denote the space of all almost periodic A~ with spectrum contained in the

functions, namely, almost periodic functions on set A, i.e. APA(~)

Note that every f E

APA(~)

= {f

E AP(~):

sp (f) C A}.

can be approximated uniformly n

A-polynomials, i.e. by exponential polynomials of type L

on~

by exponential

akeiskx, Sk E A.

k=l


ANALYTIC FUNCTIONS ON COMPACT GROUPS

301

2. Shift invariant algebras on groups. Let G _?e a compact abelian group, and letS be an additive subsemigroup of its dual r = G, containing the origin. Linear combinations over C of functions of type xa, a E S are called S -polynomials on G. Denote by As the set of all continuous functions on G whose Fourier coefficients

c~ =

J

f(g)'5t(g) da are zero for any a outside F\S. Here a is the normalized Haar

G

measure on G. The functions in As are called S-functions on G. Any S-function on G can be approximated uniformly on G by S-polynomials, and vice versa. The set As is a uniform algebra on the group G. A uniform algebra A on G is G-shift invariant if, given an f E A and g E G, the translated function f 9 (h) = f(gh) belongs to A. Every algebra of S-functions is invariant under shifts by elements of G. Vice versa, every G-shift invariant uniform algebra on G is an algebra of S-functions for some uniquely defined subsemigroup S C G (Arens, Singer [1]). Algebras As of S-functions are natural generalization of polydisc algebras E N. With G = 'JI'n, = G = zn, and = Z+., the algebra As in fact coincides with the algebra Azn = A('JI'n) on the torus 'JI'n, and Z+-functions

A('JI'n), n

r

s

+

are traces on 'JI'n of usual analytic functions in n variables in the polydisc continuous up to the boundary 'JI'n.

illln,

The maximal ideal space Ms of As is the set H(S) = Hom (S, illl), and the Shilov boundary 8As is the group G (Arens-Singer [1]). H(S) is a semigroup under the pointwise operation (cp'lj;)(a) = cp(a)'lj;(a), a E S. The Gelfand transforms j of elements f E As are continuous functions on Ms, and the space As = {j: f E As} is a uniform algebra on Ms. As shown by Arens and Singer (e.g. Gamelin [14]), As is a maximal algebra if and only if the partial order generated by the semigroup S in G is Archimedean. Note that in this case G C JR. and there is a natural embedding of the real line JR. into G so that the restrictions of S-functions on this embedding are almost periodic functions that admit analytic extension on the upper half-plane II over R Moreover, an algebra of type As is antisymmetric if and only if the semigroup S does not contain nontrivial subgroups, i.e. if S n (-S) = {0} (Arens, Singer [1]). A compact group G is said to be solenoidal, if there is an isomorphism of the group JR. of real numbers into G with a dense range. Equivalently, a compact group is solenoidal if and only if there is an isomorphism from G into R Note that the Stone-Chech compactification (3F = rd off is a solenoidal group for every additive subgroup r of R If G is a solenoidal group, then its dual group r = G is isomorphic to a subgroup of R If r is not dense in JR., then it is isomorphic to Z. In this case G is isomorphic to the unit circle 'JI', S c Z+, and therefore the elements of the algebra As can be approximated uniformly on 1I' by polynomials. Hence they can be extended on the unit disc ID as analytic functions, and therefore Ms = illl, while As ~ A(ID). If r is dense in JR., then the maximal ideal space Ms has a more complicated nature. In the case when S c JR.+ and S U ( -S) = r, the S-functions in As, are called analytic, or generalized analytic functions in the sense of Arens-Singer on G. As


T. TONEV AND S. GRIGORYAN

302

mentioned before, if S = IR+ the group G coincides with the Bohr compactification ,6IR of R In this case the maximal ideal space of the algebra AIR+ is the set IDe = ([0, 1] x G)/( {0} x G), which is called the G-disc, or big disc over G. The algebra Ar+ = Ar+ (lille) is called also the G-disc algebra, or the big disc algebra. The points in the G-disc IDe are denoted by r · g, where r E [0, 1] and g E G = ,BR We identify the points of type 0 · g, g E G, and the resulting point we denote by w. Hence, w = 0 · g for every g E G. The points of type 1 · g, g E G, we denote by g. Since lR is dense in G, the set (0, 1] x lR is dense in the G-disc IDe. Equivalently, the upper half-plane II~ (0, 1] x lR can be embedded as a dense subset of the G-disc

IDe.

Below we summarize some of the basic properties of the G-disc algebra Ar+ (lille), where r+ = r n [0, oo) (cf. Gamelin [14]). (i) Mr+ =IDe. (ii) oAr+ (lille) = G. (iii) A local maximum principle holds on Ar+ (lille), namely, for every analytic r+-function f(r ·g) on IDe, for every compact set U C IDe, and for each ro ·go E U we have lf(ro · go)l :-a if and only if b - a E P. Note that every non-negative semicharacter {! E Hom (P, [0, 1]) is monotone decreasing on P with respect to the order generated by P. Indeed, if b >- a for some a, bE r+, then b =a+ p for some pEP. Therefore, {!(b) = {!(a)(!(p) ~ {!(a) since {!(c) ~ 1 on P. Consequently, if a non-negative semicharacter {!is extendable on r+ as an element in Hom ( P, [0, 1]), then it necessarily is monotone decreasing on S C P.



305

PROPOSITION 4 (Grigoryan, Tonev [25]). A positive semicharacter e E H(S) is extendable on r+ as a positive semi character if and only if e is monotone decreasing on S with respect to the order generated by P.

Proof. Let the positive semicharacter eon S be monotone decreasing. If b E r +, then b =a- c for some a,c E S, a>- c, and 'if(b) = e(a)/e(c) is a well defined and natural homomorphic extension of eon r+. Clearly, e(a)::; e(c) if and only if 'if(b) ::; 1, i.e. if and only if 'if is a positive semicharacter on r+. THEOREM 4 [25]. Non-vanishing semicharacters
0 be a positive number. Consider the semigroup Fv = {0} u [v, oo) c R Clearly, F = Fv- Fv = lR, and F+ = JR+. Since x(a +b) = x(a)x(b) :S x(a) for every a, b E Fv, every semicharacter X on Fv is monotone decreasing. Therefore, it is extendable on JR+, namely as the characteristic function Q~+ of the origin {0}. Example 3. Let a: be an irrational number. Consider the 2-dimensional semigroup Sa = { n+mo: : n, m E Z+} c R Here the group generated by Sa is Fa = Sa-Sa= {n+mo:: n,m E Z}, while (Fa)+= FanlR+ = {n+mo:?:: 0: n,m E Z}. Clearly, Sa -=1- (Fa)+. For instance the positive number a:- [a:] E (Fa)+\ Sa. For a fixed a E (0, 1) the function 1(n + mo:) =an, n + mo: E Sa is a homomorphism from Sa to (0, 1] C Ji). However, 1 is not monotone decreasing on S. Indeed, 1(mo:) = 0, while 1(n) =an > 0 for every n >mo. The natural (and only possible) homomorphic extension 1 of 1 on (Fa)+ is given by 1(n + mo:) = an, n, m E Z,n+mo:?:: 0. However, 1 tf_ H((Fa)+), since, for instance, 1(o:- [a:])= a-[a] > 1. PROPOSITION 6 ( Grigoryan, Tonev [25]). The maximal ideal space Ms of the algebra As of analytic S-functions on G = f, F = S- S with spectrum in S c JR+ is homeomorphic to the maximal ideal space Mr+ = Ji)G of the algebra Ar+ of analytic F+-functions on G if and only if all positive semicharacters on S are monotone decreasing.

As an immediate consequence we get the following PROPOSITION 7 [25]. The maximal ideal space MAPs(lR) of the algebra APs(lR) of almost periodic functions with spectrum in a semigroup S C JR+ is homeomorphic to the G-disc Ji)G, where G = f, if and only if all positive semicharacters on S are monotone decreasing.

Since the upper half plane II = {z E C : Im z ?:: 0} can be embedded densely in the maximal ideal space M s of the algebra As (and, together, of APs (JR.)) if and only if MAs = Ji) 0 , then the upper half plane II is densely embeddable in the maximal ideal space MAPs(lR) of the algebra APs(lR) of almost periodic functions with spectrum in S if and only if all positive semicharacters on S are monotone decreasing. Note that, as shown by Boettcher [6], II is densely embedable in MAPs if and only if every additive positive function Bon Sis of type B(a) =yea for some Yo E [O,oo), or B(a) = oo, for a -=1- 0. For an a E S let VJa E H 00 be the singular function VJa(z) = eta,_z on the unit disc l!Jl. Recall that Hs is the Banach algebra on lill generated by the functions VJa(z), a E S equipped by the sup-norm on l!Jl. As mentioned in Example 1 b), Hs is a subalgebra of H 00 , which is isometrically isomorphic to the algebra As of analytic S-functions on G = (S- S)~. ·~

PROPOSITION 8 ( Grigoryan, Tonev [25]). The unit disc lill is dense in the maximal ideal space of the algebra Hs if and only if all positive semicharacters on S are monotone decreasing.

Let P be a semigroup of F that generates a partial order on F, and suppose that S c E are additive subsemigroups of P that contain the origin, and such that



307

[S]s ::J E, i.e. Nan S =1- 0 for every a E E. Then every non-negative semicharacter H(S) can be extended naturally onE as a monotone decreasing semicharacter, namely by e(a) = [e(na)pln. {} E

PROPOSITION 9 [25]. If S C E are subsemigroups of P such that E C [8] 8 , then every semicharacter c.p E H(S) on Sis uniquely extendable onE as a semicharacter in H(E), and therefore, Ms =ME·

In particular, if Sis a subsemigroup of lR such that [S]s ::J r+, then the upper half plane II is densely embedable in the maximal ideal space MAPs(IR) of the algebra APs(lR) of almost periodic functions on lR with spectrum in S. PROPOSITION 10 [25]. If Sis a subsemigroup of lR such that [S]s ::J r+, then the algebra H'; does not have corona, i.e. the unit disc]])) is dense in its maximal ideal space MH:;. PROPOSITION 11 [25]. LetS be a subsemigroup of JR. Then ME= Mr+ = ll.iia for every semigroup E with S c E c IR+ if and only if [S]s = r+, i.e. for every a E F+ = r n [0, oo) there is annE N such that na E S.

Note that under the hypotheses of this proposition, the semicharacters on all semigroups E with S C E C IR+ are uniquely extendable on r + as semicharacters on r+.

3. Automorphisms of shift invariant algebras on groups. Assume that Sn ( -S) = {0}, i.e. that S contains no non-trivial subgroups. Under this condition the algebra As is antisymmetric. An element t E Ms = H(S) is an idempotent homomorphism of S if t 2 = t. Let Is be the set of all idempotents in H(S) that are not identically equal to 0 on S. It is easy to see that Is is a subsemigroup of H(S). Clearly, an idempotent homomorphism can take values 0 or 1 only. Denote by ZL the zero set {a E S: t(a) = 0}, and by EL- the support set {a E S: t(a) = 1} of t E Is. It is easy to see that if tis an idempotent homomorphism of S, then EL is a semigroup of S, ZL is a semigroup ideal inS, ZL U EL = S, and ZL n EL = 0. PROPOSITION 12 [20]. Let As be a G-shift invariant algebra on G, where S C G. Every idempotent homomorphism t E Is possesses a representing measure supported on a subgroup of G.

Note that every idempotent homomorphism of Scan be extended uniquely to an idempotent homomorphism on the strong saturation [S]s of S, i.e. Is = I[s]. for every subsemigroup S C G. An automorphism on a shift-invariant algebra As is an isometric isomorphism c.p : As ----> As that maps As onto itself. The conjugate mapping c.p* of c.p defined by (c.p*(m)) (!) = m(c.p(f)), is a homeomorphism of the maximal ideal space Ms onto itself. For instance, the conjugate mapping c.p* of an automorphism c.p of the disc algebra A(]]))) = A;z+ is a Mobius transformation of the unit disc, i.e.

z - zo c.p*(z) = C 1 , - zoz

ICI = 1, lzol < 1.


T. TONEV AND

308

S.

GRIGORYAN

Note that if the origin 0 is a fixed point of a Mobius transformation t.p*, then ICI = 1. It is easy to see that this is also the case with the automorphisms of the subalgebra A0 (!DJ) = {f E A(!DJ) : f'(O) = 0} of the disc algebra A(!DJ), i.e. the conjugate mapping of any automorphism of the algebra A0 (!DJ) fixes the origin. Observe that the conjugate mapping of an automorphism t.p : As ---+ As maps idempotent homomorphisms of S to idempotent homomorphisms of S, i.e. 2 (f) 2 = ~(t.p(f))) t.p* :Is ---+Is. Indeed, ('P*(~)) = (~(t.p(f))) = ('P*(~))(f), i.e.

t.p*(z) = Cz for some constant C with

(t.p*(~))2

= (t.p*(~)).

An automorphism t.p of a G-shift invariant algebra As is said to be inner, if there is a T E Hom (S, S) and an element go E G such that t.p(Xa) = xa(go) · Xr(a) for every Xa E S. Every automorphism t.p of the disc algebra A(!DJ) with conjugate of type t.p*(z) = Cz, ICI = 1 is inner. Indeed, for every z E ii'ii we have (t.p(f))(z) = f(t.p*(z)) = f(Cz). For Xn E Z+ : xn(z) = zn, n ~ 0 we get ('P(Xn))(z) = ('P*(z)r = (Cz)n = cnxn(z), hence t.p(Xn) = cnxn = xn(C)xn, i.e. t.p is an inner automorphism. Arens and Singer [1] have shown that every automorphism t.p of the algebra As is inner in the case when G is a solenoidal group and S is a semigroup in JR. with

su (-S) =G.

THEOREM 5 (Grigoryan, Pankrateva, Tonev [20]). If G is a solenoidal group, then either As~ A(!DJ), or every automorphism of the algebra As is inner.

Proof. If the group S generated by S is not dense in JR., then the algebra As is a subalgebra of the disc algebra A(!DJ). If As =f. A(!DJ), then As C A 0 (!DJ). In the same way as for the algebra A0 (!DJ) one can see that in this case every automorphism is the composition by a Mobius transformation, fixing the origin, i.e. every automorphism is inner. If the group S generated by S is dense in JR., then the algebra As is a subalgebra of the S-~gebra As. If t.p is an automorphism of As then the bounded analytic function t.p(Xa)(z) does not have zeros in II. Moreover, I'P(Xa) o jl

= 1 on JR., since

lxal = 1 on G = 8As. By the Besicovitch theorem [2], ~(J(z)) = ~(z) = Ceisz = cxs(](z)), where s ~ 0, c E c, ICI = 1. It is easy to see that s E s, and

that the mapping

T:

F----+ S: Xa

f----+

X8 is a homomorphism from S to S.

4. Primary ideals of algebras of analytic functions on solenoidal groups. Characterizing various types of ideals is an important and interesting topic in uniform algebra theory. A proper ideal of an algebra is said to be a primary ideal if it is contained in only one maximal ideal of the algebra. By Ir·g below will be denoted the maximal ideal of functions in As that vanish at the point r · g E ii'ii. Recall that every primary ideal J of the disc algebra A(!DJ) which is contained in some maximal ideal of type lz0 , lzol < 1, admits the representation J = un A(!DJ), where u(z) is the unimodular function z- zo . 1- zoz



309

THEOREM 6 (Grigoryan [18]). Let F = lR and S = JR+. If J is a primary ideal of the algebra As that is contained in Iw, then either J = xs(J) Iw, or, J = xs(J) As; Every primary ideal I of As that is contained in a maximal ideal of type Ir. 9 , r · g E lDlc has a finite codimension in As.

Let Mf3 =HJ(lR) ·exp(~,BC

.

,8' :2: 0.

1

)

·

dt

1 +t 2 , ,8:2:0. Note that Mf3::) Mf3' for ,8:2:

7 [18]. Let J be a primary ideal of As that is contained in Ie Then there exists a ,8 :2: 0 such that Jj_ = (As)j_ + Cbo + Mf3. THEOREM

= Ije(D).

5. Asymptotic Almost Periodic Functions. A function f E BC(JR) is asymptotic almost periodic, if there is an almost periodic function J( x) on JR, such that limn_, 00 if(xn)- J(xn)l = 0 for every sequence {xn}~=l ---+ ±oo. Since h(x) = j(x) -1(x) E C0 (JR), we have that for every asymptotic almost periodic function f on lR there are unique E AP(JR) and hE C0 (JR) such that f = +h. One can show that

1

1

THEOREM 8. Let G = ,BJR be the the Bohr compactification ofR The maximal ideal space MAPa (JR) of the algebra of asymptotic almost periodic functions AP0 (JR) is homeomorphic to the Cartesian product G x 1!'.

Let r be an additive subgroup of JR, and let APr(JR) be the set of almost periodic F-functions. Clearly, APr(lR) EB C0 (JR) is a uniform subalgebra of AP0 (JR), containing C0 (JR). It is not hard to see that every antisymmetric subalgebra of AP0 (JR) that contains C0 (JR) is of this type. 9. Let A be an uniform subalgebra of AP0 (JR) which is invariant c JR, and a closed subalgebra A 0 of under JR-shifts. Then there is a subgroup C0 (JR), such that (a) The algebra APr(lR) of almost periodic F-functions is a closed subalgebra of A. (b) A= APr(lR) EB Ao. (c) A 0 is an ideal in A. THEOREM

f

DEFINITION

E AP0 (JR) and

r

6. A function f E BC(JR) is analytic asymptotic almost periodic if a bounded analytic extension on the upper half-plane

f possesses

II. Clearly, the set AAP0 (JR) of analytic asymptotic almost periodic functions on lR is an antisymmetric uniform algebra under the sup-norm on JR, and AAP0 (JR) C AP0 (JR). Note that AAP(JR) ~ AJR+ ~ APJR+ (JR). Consequently, MJR+ is the G-disc

lDc over the group G

=

,BR We have also the following results.

10. The maximal ideal space MAAPa(lR) of AAPo(lR) is homeomorphic to the Cartesian product lDf3JR x JD. THEOREM



310

THEOREM 11. Let G be a solenoidal group, such that its dual group r = 8 is a dense subgroup of JR, and let S be an additive subsemigroup of r+ containing the origin, with [S]s = r+. Then there is a continuous projection from MHoo onto the maximal ideal space MAAPo(IR) ~ j[j)G x j[j)_ THEOREM 12. The maximal ideal space of any subalgebra of AAP0 (JR) of type AAPs(lR) E9 B, where S c JR+ and B c Co(JR)n Hol (II), is the set MAAPs(IR)E!lB

=j[j)a x MB.

In particular, the upper half-plane II is not dense in the maximal ideal space of any subalgebra AAPs(lR) E9 B of AAP0 (JR) which contains properly AAPs(lR); The unit disc ][)) is not dense in the maximal ideal space of any subalgebra of the algebra [ea ;~i, a E S] E9 B C H 00 n A(Jl5) \ {1} ), where S is an additive semigroup in JR, and B -=1- {0} is a subalgebra of the space {! E C('ll') : f(1) = 0}.

A function f E BC(JR) is called weakly almost periodic, if the set of alllR-shifts, ft(x) = f(x + t), t E lR is relatively weakly compact in BC(JR) (e.g. Eberline [13], Burckel [7]). If W AP(JR) denotes the set of weakly almost periodic functions on JR, then AP(JR) c APo(lR) c W AP(JR). In fact, W AP(JR) = AP(JR) E9 C([-oo, oo])IIR· Similarly to Theorem 11, one can show the following THEOREM 13. The maximal ideal space MAWAP(IR) of AW AP(JR) of analytically extendable on II weakly almost periodic functions on lR is homeomorphic to the Cartesianproductll5)13IR x {([0,1] x [0,1])/([0,1] x {0})}.

The space AW AP(JR) o (T,n)EJ

T(T,n)

n,

'

= {(n!)- 1 (ml'Yl + m2'Y2 + · · · + mk'Yk)

: mj E Z, j

=

1, ... k} ~ zk.

Let P(T,n) = To, n) + = r(T,n) U [0, oo). If Aph,nl is the algebra of analytic P(T,n)functions on G, one can show that AIR+ = [ ~ Ap(-y,n) (lDla)]. A similar expression (T,n)EJ

holds for the algebra As, S c JR+. Uniform algebras that can be expressed as inductive limits of disc algebras A(lDl) are of special interest. Consider the inverse sequence {Jl5)k+l, r:+l }k'=l, j[j)k = Jl5) and r:+ 1 (z) = zdk on j[j)k· The limit lim {Jl5)k+ 1 ,r:+ 1 } of the inverse sequence f--

k-+oo

{Jl5)k+l, r:+l }, is the GA-disc j[j)GA = ([0, 1] x GA)/( {0} x GA) over the group GA = fA. There arises a conjugate inductive sequence {A(Jl5)k), i~+l }f of algebras A(Jl5)) ~



A('JI') with connecting homomorphisms i~+l

: A(IDJk)

= (f(z))dk, i.e.

(i~+l(f))(z)

i~+l

-+

311

A(IDJk+l) defined by

= (Tt+ 1)*.

The elements of the component algebras A(IDJk) can be interpreted as continuous functions on G A· The uniform closure A(IDJcA) = [ lim { A(IDJk), i~+l }] in C(IDJcA) of ---+ k-->oo

the inductive limit of the system { A(IDJk), i~+l }k"= 1 and the corresponding restricted algebra [lim {A('ll'k),i~+l}] are isometrically isomorphic to the GA-disc algebra ---+ k-->oo

ArA+' i.e., to the algebra of analytic rA+-functions on the GA-disc (e.g. [21]).

Consider an inductive sequence of disc algebras

where the connecting homomorphisms i~+l with Mi~+'(A('ll'k)) = IDJ and 8(i~+l(A('ll'k)))

Bk · ]]]) .

-+ ]]])

'

Bk(z) =

eilh

21(k) link ( z- -(k)

: A('JI'k)

-+

A('JI'k+l) are embeddings

= 1!'. There are finite Blaschke products

)

'

lz(k) I < 1 such that ik+l = B* for l ' kk

l=l 1- zl z every k E N, i.e. i~+l(f) = f o Bk· Let B = {Bk}k=l be the sequence of finite Blaschke products corresponding to i~+l, i.e. (Bk)*(z) = i~+ 1 (z). Let A= {dk}k=l be the sequence of orders of Blaschke products {Bk}k=l and let FA c iQ be the group generated by 1/mk, k = 0, 1, 2, ... , where mk = IT~=l dt, m 0 = 1. Consider the inverse sequence

IDJ1 ~

IDJ2 .,_1}.2. IDJ3 ~

IDJ4 ~

···

+----

DB.

The inverse limit DB = lim ....._ {IDJk+ 1, Bk} is a Hausdorff compact space. The limit k-+oo

of the composition system { A(IDJk), ,6~+ 1 }1 of disc algebras A(IDJk) and connecting k+l k+l homomorphisms ,6k = BJ:, : A(IDk)-+ A(IDk+l): (,6k (f))(zk+l) = f(Bk(ZkH)) is an algebra of functions on DB whose closure [lim {A(IDJk),,6~+ ---+ k-->oo

1

}]

= A(DB)

in C(DB) we call a Blaschke inductive limit algebra. It is isometrically isomorphic to the algebra [ lim { A('JI'k), ,a~+l} ]. ---+ k-->oo

PROPOSITION 13 (Grigoryan, Tonev [21]). Let B = {Bk}k=l be a sequence of finite Blaschke products and let A(DB) = [lim {A(IDk), Bk}] be the corresponding ---+ k-->oo

inductive limit of disc algebras. Then (i) A(DB) is a uniform algebra on the compact set DB= lim ....._ {IDJk+ 1, Bk} . (ii) The maximal ideal space of A(DB) is DB. (iii) A(DB) is a Dirichlet algebra. (iv) A(DB) is a maximal algebra.


k-->oo

T. TONEY AND S. GRIGORYAN

312

(v) The Shilov boundary of A(D 8

)

is a group isomorphic toGA, and its dual

group is isometric to the group FA

00

U (1/mk)Z

~

k=O

C Ql, where

mk

=

THEOREM 14 [21]. ~et G be a solenoidal group, i.e. G is a compact abelian group with dual group G isomorphic to a subgroup r of R The G-disc algebra Ar+ is a Blaschke inductive limit of disc algebras if and only if r is isomorphic to a subgroup of Ql. THEOREM 15 [21]. Let B = {Bk}k"= 1 be a sequence of finite Blaschke products on ID, each with at most one singular points z6k) and such that Bk(zbk+ 1 )) = z6k). Then the algebra A(DB) is isometrically isomorphic to the algebra A(r,t)+ with A= {dk}k"= 1 , where dk = ordBk. In particular, if every Blaschke product Bk in the above theorem is a Mobius transformation, then the algebra A(D8 ) is isometrically isomorphic to the disc algebra Az = A('ll'). 2. Inductive limits of algebras on subsets of G-discs. Let lDJ[r, 11 = {z E C : r :::= jzj :::= 1}, and blDJ[r, 11 = {z E C : jzj = r or jzj = 1} is the topological boundary of JD)[r, 11. Denote by A(lDJ[r, 11) the uniform algebra of continuous functions on the set l!][r, 1 ] that are analytic in its interior. Note that A(lDJ[r, 11) = R(lDJ[r, 11), the algebra of continuous rational functions on JD)[r, 11. By a well known result of Bishop, the Shilov boundary of A(lDJ[r, 1]) is blDJ[r, 11, and the restriction of A(lDJ[r, 11) on blDJ[r, 1] is a maximal algebra with codim (Re (A(lDJ[r, 11) jb[JIIr,lJ)) = 1. These results can be extended to the case of analytic r+-functions on solenoid groups (e.g. S. Grigoryan [19]). Let G be a solenoidal group, and its dual group is denoted as r c R Let JD)~' 1 1 = [r, 1] X G, 0 < r < 1 be the [r, 1]-annulus in the G-disc IDe, and let A(lDJ~' 1 1) = R(lDJ~' 1 1) be the G-annulus algebra on lDJ~' 1 1, generated by the functions Xa, a E T. Let A= {dk}k"= 1 be a sequence of natural numbers and T;+ 1 (z) = zdk, and let r be a fixed number, 0 < r :::= 1. For every k E N consider the sets Ek where

-[rlfmk 1]

= lDJ mk

'

= {z

E

-

lDJ: r

1/

mk

:::= jzj :::= 1}

= f1~= 1 dl, m 0 = 1, and E 1 = -lDJ[r, 1] ~

E2 ~

= h o T 2 o · · · o Tk_ 1 )- (lDJ ' ),

jj][r, 1 ].

E3 ~

2

3

k

1 -[r 1]

There arises an inverse sequence E 4 ;--I1_

...

of compact subsets of lDJ. Consider the conjugate composition inductive sequence .• 0' where the embedings i~+ 1 : A(Ek) ---+ A(Ek+l) are the conjugates of zdk, namely, (i~+l of) (z) = f (zdk). Let G A denote the compact abelian group whose dual group



rA

=

GA

313

is the subgroup of Ql generated by A. The algebra [lim {A(Ek), i~+l }] -----t

k-+oo

is isomorphic to A(lDlS' 1l).

THEOREM 16 [21]. LetFn+l =B;; 1 (Fn), F1 =lDlS' 1l. IftheBlaschkeproducts Bn do not have singular points on the sets Fn for any n E N, then 'D~' 11 ~ lDlS' 11 , and the algebra A('D~' 1 ]) = [lim {A(Fn), B~}] is isometrically isomorphic to the -----t

G-annulus algebra A(lDlS' 1]).

n-+oo

Below we summarize some of the basic properties of the algebra A('D~' 1 l) (see [21]). (a) The maximal ideal space of A('D~' 1 ]) is homeomorphic to the set lDlS' 1l. (b) The Shilov boundary of A('D~' 11 ) is the set blDlS' 1] = {r, 1} x G. (c) A('D~'

1 l)

(d) codim(Re(A('D~'

is a maximal algebra on its Shilov boundary. 11

)1bllllr,1J)G)

= 1.

Let B = {B 1, B 2 , ... , Bn, ... } be a sequence of finite Blaschke products on~ and let 0 < r < 1. Let Dn+l = B;; 1(Dn), D1 = ][J)[O,r] = {z E lDl: lzl::; r}. Consider ][J)[o,r]

~

D

2

~

D

3

~

of subsets of lDl. The inductive limit A('D~,r]) algebra on its maximal ideal space ~

D

4

~

· · ·

'D[O,r] B

= [lim {A(Dn), B~}] -----t

is a uniform

n-+oo

{Dn, Bn-11Dn} = 'D~,r]

C 'DB.

k->oo

PROPOSITION 14 [21]. Let B = {B1, B 2 , B 3 , ... } be a sequence of finite Blaschke products on ~ and let 0 < r < 1. Suppose that the set Dn does not contain singular points of Bn-1 for every n EN. Then (i) There is a compact set Y such that 'D~,r] = ~ {Dn+l• BniDn+,}

M A(v~·rl)

n-+oo

is homeomorphic to the Cartesian product ][J)[O,r] x Y.

(ii) A('D~,r]) is isometrically isomorphic to an algebra of functions f(x, y) E C(][J)[O,r] x Y), such that f( ·, y) E A(][J)[O,r]) for every y E Y. (iii) A('D~,r])IIJliD,rlx{y} ~ A(lDl[O,r]) for every y E Y. The proof makes use of the fact that every finite Blaschke product of order n generates an n-sheeted covering over any simply connected domain V c ][]) free of singular points of B. Proposition 14 implies that the one-point Gleason parts of the algebra A('D~,r]) are the points of the Shilov boundary b 'D~,r] ~ 'II' r x Y. PROPOSITION 15 [21]. Let B = {B1, B 2 , B 3 , .•. } be a sequence of finite Blaschke products on ~. and let 0 < r < 1. Suppose that (a) For every n EN the points of the set :F = (B 1 o B 2 o · · · o Bn_I)- 1(0) are the only singular points for Bn_ 1 in Dn (b) All points in (a) have one and the same order dn_ 1 > 1.


314


Then (i) There is a compact y such that v~,r]

= ~

{JD)n+1, BniDn+l}

= M A(D~,r])

k-+oo

is homeomorphic to the Cartesian product JD)~~l x Y, where A= {dk}k'= 1 is the sequence of the orders of Bk. (ii) The algebra A(V~,r]) on 1)~,r] is isometrically isomorphic to an algebra of functions f(x, y) E C(JD)~~l x Y), such that J(-, y) E A(JD)~~l) for every yEY.

(iii) A(D~,r])I[JIIO,rJx{y}

=

A(JD)~~l)

for every y E Y.

The set Yin Propositions 14 and 15 is homeomorphic to the set { {Yn}~= 1 , Yn E (B1 o B 2 o · · · o Bn- 1)- 1 (0)}. Proposition 15 implies that there are no single-point

Gleason parts of the algebra A(D~,r])

within the set M A(D~,rJ)

\ bV~,r]

U {w}

X

Y,

where w is the origin of the G A-disc ll)GA. As an immediate consequence we obtain that A(D~,r]) is isomorphic to a Gdisc algebra if and only if the set Y consists of one point. In particular, in the above setting the algebra A(D~,r]) is isomorphic to a G-disc algebra if and only if every Blaschke product Bn has a single singular point Zbn) in vtl such that Bn(zbn)) = Zbn+ 1 ) for all n big enough. 3. Gleason parts of inductive limits of disc algebras on G-discs. The celebrated theorem by Wermer [36] asserts that in every non-single-point Gleason part of the maximal ideal space of a Dirichlet algebra can be embedded an analytic disc. Therefore it is of particular interest to locate single-point Gleason parts of an algebra, and especially those of them that do not belong to the Shilov boundary. While every point in the Shilov boundary is a separate Gleason part (e.g. Gamelin [14]), the opposite is not always true, i.e. there are single-point Gleason parts outside the Shilov boundary. For instance, if G is a solenoid group with a dense in lR dual group, then the origin w = ({0} x G)/({0} x G) E JD)G of the G-disc JD)G is a single-point Gleason part for the G-disc algebra Ar+. Of course w rf. 8 Ar+ =G. Given a sequence of Blaschke products B = {Bn};:"= 1 on ll), consider the Blaschke inductive limit algebra A(VB) = [lim {A(ll)k), Bk}] on the compact ---+

set VB

TB

=

=

k-+oo

lim {ll)k, Bk-d· Recall that the Shilov boundary of A(VB) is the group

1 then there is only one single-point Gleason part in the set VB\ TB. The proof of Theorem 18 involves a thorough study of one-point Gleason parts of the algebras involved.

4. Inductive limits of H 00 -spaces on G-discs. Let I = { iZ+ 1}k'= 1 be a sequence of homomorphisms iZ+ 1 : H 00 (]]J)) ----> H 00 (]]J)). Consider the inductive sequence H

00

(]]J)1)

·2

~

·3

H 00 (]]J)2) ~

H

00

(]]J)3)

·4

~

of algebras H 00 (]]J)k) ~ H 00 (]]J)). Every adjoint mapping (iZ+ 1)* : Mk ..- Mk+l maps the maximal ideal space Mk+ 1 of H 00 (]]J)k+l) into the maximal ideal space Mk of H 00 (]]J)k)· The inverse limit

M1

(;2)*

~ 1 -M2

(;3)*

~ 2 -M3

r;4)*

~ 3 -

M4

(;5)*

~ 4 -

••• .._____

'Dr

is the maximal ideal space of the inductive limit algebra H 00 ('Dr)

=

[lim {H 00 (]]J)k),iZ+l}]. --+ k-+oo

Recall that the open unit disc ]]J) is a dense subset of every Mk· In general, the mappings (iZ+ 1)* are not obliged to map ]]J)k+l onto itself. The most interesting situations, though, are when they do. Here we suppose that the mappings (iZ+ 1)* are inner non-constant functions on ]]J). For instance, algebras of type H 00 ('Dr) are the algebras [lim {H 00 (]]J)k), (zdk)*}dkEA] = H 00 (VA) C H 00 (]]J)aA), and also --+ k-+oo

the algebras of type H 00 (VB)

= [ ~ {H (]]J)k), Bk}], where B = {Bk}k'= 1 is 00

k-+oo

a sequence of finite Blaschke products Bk : ]]J) ----> ]]J). Note that H 00 (VB) is a commutative Banach algebra of functions on VB. Let A= {dk}k'= 1 be the sequence of orders of Blaschke products {Bk}k'= 1 from the mentioned above example, and let C IQ be the group generated by 1/mk, k = 0, 1, 2, ... ' where mk = rr~=1 dz, mo = 1.

rA

THEOREM 19 [21]. Let B = {Bk}k'= 1 be a sequence of finite Blaschke products on iiSi, each with at most one singular points zbk)' and such that Bk(zbk+ 1 )) = zbk). Then the algebra H 00 (VB) is isometrically isomorphic to the algebra H 00 (VA) for A= {dk}k'= 1 with dk = ordBk· For instance, if the Blaschke products Bk are of type Bk (z) = zdk r.pk( z), where 'Pk are Mobius transformations and dk > 1, then the algebra H 00 (VB) is isometrically isomorphic to the algebra H 00 (VA), where A= {1/dk}~ 1 . If every Blaschke



316

product Bk in Theorem 19 is a Mobius transformation, then the algebra H 00 (VB) is isometrically isomorphic to the algebra H 00 . Indeed, the last theorem implies that H 00 (VB) ~ H 00 (VA) with A= {1, 1, ... }. Therefore rA = z and GA = 1!'. Let P = {cp 1 , cp 2 , ... , 'Pk, ... } be a sequence of non-constant inner functions on lDl. Consider the inverse sequence ]]])1

where lDlk

~

+---'£.1.

]]])2 ~

]]])3 ~

]]])4 ~

...

lDl. Denote by V
0

on lDl, there exist functions g1, ... , gk in H 00 such that ~~= 1 fJgj = 1 on lDl; If llfjlloo ~ 1, then gj can be chosen to satisfy the estimates llgjll ~ C(k,O") for some constant C(k, O") > 0. Next theorem is the corona problem for the algebra H 00 ('D). THEOREM 20 (Grigoryan, Tonev [21]). Ifh,h ... ,fn, II!JII ~ 1, aret:P-hyperanalytic functions on V
0 for each x E 'D, then there is a constant K ( n, o) and t:P-hyper-analytic functions g 1 , ... , gn on V with llgjll ~ K(n, o), such that the equality h(x)g1(x) + · · ·+ fn(x)gn(x) = 1 holds for every point x in the set V. In the case when t:P = { z 2 , z 3 , ... , zn+l ... } the corresponding ~et V
1 gEG



317

by 1-{ 00 (][)) 0 ) the weak*-closure of Ar+ in L 00 (G,a) (cf. Gamelin [14]). Clearly H 00 (][)) 0 ) is a closed subalgebra of 1-{ 00 (][))c). Let I be a directed set. We consider a family {ri}iEI of subgroups of r indexed by I, such that Ti 1 c Ti 2 whenever i1--< i2. Let r = limTi, and Hi=(][))c) denotes --+

iEI

the space of functions f E H (][))c) with sp (f) c ri. The family {Hp(][))c)}iEI of subalgebras in H= (][))c) is ordered by inclusion. Denote by H[ (][))c) the closure of the set U Hi=(][))c) = limHi00 (][))c) with respect to the norm I · lloo· H[(][))c) 00

i"EJ

iEI

is the set of !-hyper-analytic functions on ][))G· In a similar way we define the algebra 1-f[(][))c) as the I · 11 00 -closure of the inductive limit lim 1-fr'(][))c), where --+

HT, (][))c)

= {f E

7-(00

iEI

(][))G) : sp (f) c ri}.

'

THEOREM 21 (Grigoryan, Tonev [22]). Let G be a solenoidal group such that its dual group r = G is the inductive limit of a family {ri};EI of subgroups of r, i.e. T = lim Ti. Let Hp (][))c) and Hp (][))c) be the spaces offunctions in H 00 (][))c) --+

'

iEI

'

[resp. in 1-f=(][))c)] with spectra in ri, i E I. Then the following statements are equivalent. (a) H 00 (][))c) = H[ (][))c) and 1-f= (][))c) = 1-f[ (][))c). (b) Hoo (][))c) = U HF, (][))c) and 1i 00 (][))c) = U HP, (][))c). iEI

(c) Every countable subgroup T 0 in family {Ti};EI.

r

iEI

is contained in some group from the

Example 4. Let r = Q be the group of rational numbers with the discrete topology. Assume that {ri};EI is an inductive system of subgroups of Q such that Q = lim ri. The last theorem implies that if Q itself is not one of the groups in the --+

iEI

family { ri hE I' then H[ (][))c) =1- H 00 (][))G). In the case when all subgroups T;, i E I are isomorphic to ::Z, the algebra H[ (][))c) coincides with the algebra of hyper-analytic functions (e.g. [34]). As seen above, in this case the space H[ (][))c) differs from H 00 (][)) 0 ). The properties of subalgebras of H 00 (][)) 0 ) on general compact groups G are less known. In particular it is not known if they possess a corona, and their maximal ideal spaces and Shilov boundaries lack a good description. Example 5. Let r = lR and let A C JR+ be a basis in lR over the field Q of rational numbers. Then lR = ~ rb,n), where ('y,n)EJ

Given an (r, n) E J, consider the set



318

U Hh,n)(JfJJa) under the II· 11 00 -norm, i.e. the (r,n)EJ Hh,n)(JfJJa) is a subalgebra of H 00 (JfJJa). There arises

The closure H'J(JfJJa) of the set inductive limit algebra

!!s

('y,n)EJ the question of whether or not the algebra H'J(JfJJa) coincides with H 00 (JfJJa).

THEOREM 22 (Grigoryan, Tonev [22]). The set HJ(JfJJa)

is a proper closed subalgebra of H 00 (JfJJa).

=

lim H(ooJ,n )(JfJJa) ------t (r,n)EJ

Proof. The inclusion H'J(JfJJa) C H 00 (JfJJa) is easy (e.g. [12]). Assume that Hh,n)(JfJJa). By the previous theorem, the countable ('y,n)EJ subgroup Q C IRis a group in the family {T(r,n)}('Y,n)EJ, which is impossible since Th, n) is isomorphic to 71} for some k E N.

H 00 (JfJJa) = HJ(JfJJa) =

!!s

The algebra H 00 (JfJJa) is isometrically isomorphic to the algebra HAPr+ (IR) (IR) C

H (IR) of boundary values of almost periodic r +-functions on IR that are analytic in the upper half plane. Similarly, the algebra H'J(JfJJa) is isomorphic to a subalgebra H'J (IR) of HAPr+ (IR) (IR). As the last theorem shows, these algebras are different. 00

Algebras of type H[ (JfJJa) were introduced in connection with the corona problem for algebras of analytic T-functions (Tonev [32]). R. Curto, P. Muhly and J. Xia [12] have introduced similar algebras of this type in connection with their study of Wiener-Hop£ operators with almost periodic symbols.

6. Bourgain algebras and inductive limits of algebras. Bourgain algebras of a Banach space were introduced in 1987 by J. Cima and R. Timoney [9] as a natural extension of a construction due to J. Bourgain [5]. The concept of tightness of an algebra was introduced by Cole and Gamelin [10]. Let A C B be two commutative Banach algebras, and 1r A : B - t B /A is the natural projection. The mapping Sf: A-t (f A+A)/A c B/A; Sf: g f----> nA(fg) is called the Hankel type operator. DEFINITION 7. An element fEB is said to be (a) a Bourgain element, (b) a we-element, (c) a c-element for A, if the Hankel type operator Sf : A-t B/A is correspondingly (a) completely continuous, (b) weakly compact, (c) compact. The Bourgain algebra of A relative to B is the space Af of all Bourgain elements for A in B [9]. PROPOSITION 17 [35]. If the range Sf(A) = 7rA(f A) of the Hankel type operator Sf for an f E B is finite dimensional then f E Af In particular, (As)f(G) = C(G) if As is a maximal algebra and xS \ x is a finite set for a character X E G\S. Indeed, X E (As)f(G) by the above proposition. Since x ~ S, then x ~As, and consequently (As)f(G) = C(G) by the maximality of A.



Example 6. Let H be a finite group, G = (H EB = C(G).

(As)f(G)

Z'.f and

S

319

~

H EB Z+. Then

Note that if r- r = G and xS \X is finite for every X E r then every character G has finitely many predecessors in r. As it follows from Proposition 17, (Ar)f(G) = C(G), and therefore the corresponding big disc algebra Ar possesses the Dunford-Pettis property. X E

THEOREM 23 (Yale, Tonev) [35]. Let G = j)JR. be the Bohr compactification of R The Bourgain algebra (AlR'.+)f(G) of the big disc algebra Affi'.+ is isomorphic to Affi'.+· Proof. Clearly, JR. is a subset of (AlR'.+ )f(G). Since, as one can easily see, the

sequence of real valued functions

os n

~

i.pn ( x)

1 + ei:;'; 12n converges pointwisely to 1 2

= 1

oo foe evecy x E Ill!, then the ceal wJ ned functione ¢,(g)

~

II + ~ * I'" (g)

converge pointwisely to 1 as n-+ oo for every g E G. Suppose that x3 E (AlR'.+ )f(G). Consider the sequence ~n(g) = ?/Jn(g)-1, where 1/!n is as before. {x 1 ~n}n is a weakly null sequence in Affi'.+ since it converges pointwisely to 0 on G. By the Bourgain algebra property there are functions hn E Affi'.+ such that llx 3 x 1 ~n- hnll < 1/n for every n, where I · I is the sup norm on G. By integrating over Ker(x~ ), if necessary, we can assume that hn = qn

then (x'¢,)(g)

~

(X~)

rc r

2 for some polynomial qn. If Pn (z) = ( 1 ; z) n,

( x* (g))" ( 1+ ;*(g)

+ ~*

~ p,(x* (g)).

(.q)

Foe j

~

So+ S1 + · · · + S1 . of Pn, where Sk is the k-th J+1 partial sum of Pn, we have 4n(2n + 1)0"~~(z) = (1 + z) 2 n + 2n(1 + z) 2 n-l = (2n + 1 + z)(1 + z) 2 n-l. Now 2n the j-th Cesaro mean

O"jn

IIX 3 X 1 ~n= ma0x I(X gE

= TlsiPn

(X~

1

~n)(g)-

=

hnll

=

ma0x I(X gE

3

X 1 ~n)(g)-

hn(g)l

(x 3 hn)(g)l =max l(x 1 ?/Jn)(g)- X1 (g)- X3 (g)hn(g)l gEG

(g)) - X1 (g)-

(X~

3

(g)) n qn

(X~

(g)) I =

~~f

1Pn(z)-zn-z 3 nqn(z)l.

3 nqn(z)(z) Note that O"~~(z)-zn (z) = O"~~(z)-zn-z because the Cesaro mean 0"2n depends only on the first 2n terms of the Taylor series. Since the inequality max 10"~ (z) I

::; max lf(z)l holds for every zE'li'

f E A('f) we see that


zE'li'


320

However, CJ~~(z)-zn

(z) = CJ~~(z\z)-

1/2 as n-+ oo for odd n. Hence AJR+ by the maximality of AJR+.

x

3

zn(n + 1)/(2n+ 1) and thus CJ~~(z)-zn ( -1)-+ ~ (AJR+)f(G) and consequently (AJR+)f(G) =

THEOREM 24 (Tonev [33]). Let {Acr }crEE, {Bcr }crEE be two monotone increasing families of closed subspaces of a commutative Banach algebra !3 such that Bcr are algebras, and Acr C Bcr for every CJ E E. Let A = [ U Acr] be a (linear) sub-

space, and let B

= [ U Bcr] be crEE

crEE a subalgebra of !3. Suppose that for every CJ E E

there is a bounded linear mapping r"" : B -+ Bcr, such that (i) rcriBa = idBa (ii) rcr(fg) = frcr(g) for every f E Bcr, g E B

(iii) sup llrcrll < oo. crEE ThenAr c [ U (Acr)~a]. crEE Proof. Let f E B be a Bourgain element for A. Fix a CJ E E, and consider a weakly null sequence { IPn} in Acr C A. Then { IPn} is also a weakly null sequence in A since FIAa E A; for any FE A*. Therefore, one can find hn E A such that llfiPn- hnll-+ 0. Hence,

llrcr(f)ipn- rcr(hn)l[

= llrcrUIPn)- rcr(hn)[[:::;

l[rcr[[[[fiPn- hnll-+ 0.

Consequently, r"" (f) is a Bourgain element for Acr, i.e. r cr (f) E ( Acr) ~" for every CJ E E. Note that under the hypotheses every f E B is approximable by the elements rcr(f) in the norm of !3. Indeed, let fern E Bern be such that fern -+f. Then llf-rcrn(f)ll:::; llf-fcrJ[+IIrcrn(fcrJ-rcrn(f)ll:::; llf-fcrJ[+supllrcrJIIIfcrn -fll· Hence rcrn (f) -+ f and, consequently, f E [ U (Acr )~""]. crEE

r =

lim Fer, let H'f?a = {f E H 00 (l!J)c) : sp (f) c Fer}. Note that HF_" is a crEE closed subalgebra of H 00 (l!J)c), and H~ c HT,. if and only if Fer c Tr. Therefore, the family { H~} crEE of subalgebras of H 00 (l!J)c) is ordered by the inclusion. Denote by H'f? the closure of the set U H'f? with respect to the norm II · lloo· Theorem If

~

+

aEE

a

lim r"" and G = f, then the Bourgain elements for Hroo are ~ + crEE approximable in the L 00 -norm on G by Bourgain elements for H~, CJ E E. Note that HT?r, H 00 (l!J)c), and the weak* closure H 00 (G, dCJ) of Ar+ in L 00 (G, dCJ) are commutative Banach subalgebras of L 00 (G,dCJ), which are in principle different from each other, except in the case of G = 'f, when they coincide (Grigoryan [19]). 24 implies that if

r =

The algebra HQ., 1n = H 00 o Xl/n = {f o Xl/n : f E H 00 } is a closed subalgebra of H 00 (l!J)c). The closure H~ of U H~ with respect to the norm II · lloo is the nEN algebra of hyper-analytic functions on G = jJQ ( cf. Tonev [34]). By Theorem 24 the Bourgain algebra of H~ is contained in the algebra H~ + C(G). ~+

~1/n



321

THEOREM 25 (Tonev [33]). If the hypotheses of Theorem 24 are satisfied, then A~c c [ U (A,,)~~]; A~ c [ U (Au)~"]; crEE

(H'f',Jt;(c) C [ U (HT::)t;(G)]; crEE

In particular, the algebra

(HT',_)~oo(G)

crEE

C [ U (HT::)~oo(G)]. crEE

H?ft_ + C(;JQ) contains the spaces (H?ft_)t;(f3Q) and

(H?ft_) ~ oo (f3Q). A uniform algebra A C C (X) is said to be tight [strongly tight] if

= C(X) every f E C(X) is a we-element [resp. c-element] for A, i.e. if (A)~~G) [resp. (A)f(G) = C(X)] (cf. Cole, Gamelin [10], also Saccone [30]). Theorem 25 implies that the algebra H?ft_ is neither tight nor strongly tight. References 1. R. Arens and I. Singer, Generalized analytic functions, Trans. Amer. Math. Soc., 81(1956), 379-393. 2. A. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932. 3. S. Bochner, Boundary values of analytic functions in several variables and of almost periodic functions, Ann. Math., 45(1944), 708-722. 4. H. Bohr, Zur Theorie der fastperiodischen Funktionen, III. Dirichletentwicklung analytischer Funktionen, Acta Math., 47(1926), 237-281. 5. J. Bourgain, The Dunford-Pettis property for the ball algebra, polydiscalgebra and the Sobolev spaces, Studia Math., 77(1984), 245-253. 6. A. Boettcher, On the corona theorem for almost periodic functions, Integral Equations and Operator Theory, 33(1999), 253-272. 7. R. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, 1970. 8. L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76(1962), 542-559. 9. J. Cima and R. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Michigan Math. J. 34(1987), 66-104. 10. B. Cole and T. W. Gamelin, Tight uniform algebras, J. Funct. Anal. 46(1982)' 158-220. 11. C. Corduneanu, Almost Periodic Functions, Interscience, N.Y., 1968. 12. R. Curto, P. Muhly and J. Xia, Wiener-Hopf operators and generalized analytic functions, Integral Equations and Operator Theory, 8(1985), 650-673. 13. W.F. Eberline, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., 67(1949), 217-240. 14. T.W. Gamelin, Uniform Algebras, 2nd ed., Chelsea, New York, 1984. 15. I. Glicksberg, Maximal algebras and a theorem of Rad6, Pacific J. Math. 14(1964), 919-941. 16. S. Grigoryan, Algebras of finite type on a compact group G, Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 14(1979), No. 3, 168-183. 17. S. A. Grigorian, Maximal algebras of generalized analytic functions (Russian), Izv. Akad. Nauk Armyan. SSR Ser. Mat. 16(1981), no. 5, 358-365.



322

18. S. A. Grigoryan, Primary ideals of algebras of generalized analytic functions, J. Contemp. Math. Anal. 34, no. 3(1999), 26-43. 19. S. A. Grigorian, Generalized analytic functions, Uspekhi Mat. Nauk 49(1994), 3-42. 20. S. Grigoryan, T. Ponkrateva, and T. Tonev, Inner Automorphisms of Shiftinvariant Algebras on Compact Groups, J. Contemp. Math. Anal., Armen. Acad. Sci., 5(1999), 57-62. 21. S. Grigoryan and T. Tonev, Blaschke inductive limits of uniform algebras, International J. Math. and Math. Sci., 27, No. 10(2001), 599-620. 22. S. A. Grigoryan and T. V. Tonev, Inductive limits of algebras of generalized analytic functions, Michigan Math. J., 42(1995), 613-619. 23. S. A. Grigoryan T. Ponkrateva, and T.V. Tonev, The validity range of two complex analysis theorems, Compl. Variables, (2002). 24. S. Grigoryan and T. Tonev, Inductive limits of algebras of generalized analytic functions, Michigan Math. J., 42(1995), 613-619. 25. S. A. Grigoryan and T. V. Tonev, Linear multiplicative functionals of algebras of S -analytic functions on groups, Lobachevsky Math. J., 9(2001), 29-35. 26. B. Jessen, Uber die Nullstellen einer analytischen fastperiodischen Funktion. Eine Verallgemeinerung der Jensenschen Forme!, Math. Ann. 108(1933), 485-516. 27. E. R. van Kampen, On almost periodic functions of constant absolute values, J. Lond. Math. Soc. XII, No. 1(1937), 3-6. 28. G. M. Leibowitz, Lectures on Complex Function Algebras, Scott, Foresman and Co., Glenview, IL, 1970. 29. L. Loomis, Introduction in Abstract Harmonic Analysis, Van Nostrand, Princeton, N.J., 1953. 30. S. Saccone, Banach space properties of strongly tight uniform algebras, Studia Math., 114(1995), 159-180. 31. A. Sherstnev, An analog of the Hahn-Banach theorem for commutative semigroups, preprint. 32. T. Tonev, The Banach algebra of bounded hyper-analytic functions on the big disc has no corona, Analytic Functions, Lect. Notes in Math., Springer Verlag, 798(1980), 435-438. 33. T. Tonev, Bourgain algebras and inductive limit algebras, In: Function Spaces (Ed. K. Jarosz), Contemporary Math., AMS, 232(1999), 339-344. 34. T. Tonev, Big-planes, Boundaries and Function Algebras, Elsevier~ NorthHolland, 1992. 35. K. Yale and T. Tonev, Bourgain algebras and the big disc algebra, Rocky Mountain J. Math., preprint. 36. J. Wermer, Dirichlet algebras, Duke Math. J., 27(1960), 373-382.

DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF MONTANA, MISSOULA, MONTANA

59812-1032

CHEBOTAREV INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN, SIA


Rus-

This volume presents papers from the Fourth Conference on Function Spaces. The lectures covered a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), LP-spaces, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces, and related subjects. Included are 26 articles written by leading experts. Known results, open problems, and new discoveries are featured. Most papers are written for nonexperts, so the book can serve as a good introduction to the material presented.

ISBN 0-8218 - 3269- 7

9 780821 832691

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