{f : supjâJ âÏâΩj m(Ï)|f(Ï)|2 < 1} for a suitable partition {Ωj : j â J} of the. 1 ... extreme points of the dual unit ball of the associated JB*-triple as certain finite.
Symmetric continuous Reinhardt domains ´ and B. ZALAR L.L. STACHO
1. Introduction A classical Reinhardt domain is an open connected subset D of C n the space of all complex n -tuples satisfying (z1 , . . . , zn ) ∈ D ⇒ (eit1 z1 , . . . , eitn zn ) ∈ D for all real n -tuples (t1 , . . . , tn ) ∈ IRn . A Reinhardt domain is called complete if (z1 , . . . , zn ) ∈ D ⇒ (ζ1 z1 , . . . , ζn zn ) ∈ D whenever |ζ1 |, . . . , |ζn | ≤ 1 . In 1974 [11] Sunada investigated the structure of bounded Reinhardt domains containing the origin from the viewpoint of biholomorphic equivalence. He was able to describe completely the symmetric Reihardt domains which, up to linear isomomorphism, turned to be direct products of Euclidean balls. Our aim in this paper is to study infinite dimensional analogs of symmetric Reihardt domains. There are several ways of extending the definition of classical Reihardt domains. We are however motivated by a recent interesting work of Vigu´e [12] who considered continuous products of discs (of different radius). He obtained a rather surprising result that such a continuous product is not symmetric automatically. We intend to give a definition of a continuous Reinhardt domain which includes the domains studied by Vigu´e in such a way that our results can also lead to some continuous mixing of Euclidean balls generalizing the work of Sunada. In the classical definition of a Reinhardt domain n -tuples can be viewed as (continuous) functions from the discrete space Ω := {1, . . . , n} to C . In this terminology, complete Reinhardt domains are closely related with the ordering f ≥ 0 ⇐⇒ f (ω) ≥ 0 for all ω ∈ Ω . Namely D ⊂ C(Ω) is a complete Reinhardt domain if f ∈ D and |g| ≤ |f | implies g ∈ D . This definition can simply be extended to any complex function lattice. In the setting of bounded continuous functions, the Gel’fand-Naimark theorem naturally leads to the lattice C0 (Ω) of all continuous functions over a locally compact Hausdorff space vanishing at infinity. We note that in 1978 Braun-Kaup-Upmeier [1] introduced another interesting generalization of Reinhardt domains in infinite dimensional setting. This concept is different from ours and even in their later works their school changed the terminology so that today their concept is called bicircular domains. In the present paper we concentrate on the case of symmetric domains. We prove that continuous symmetric Reinhardt domains are in certain sense made of Euclidean balls but in a more complicated manner than simple direct products. Moreover, for a given domain D , the dimensions of these balls are simultaneously bounded by a finite constant determined by some geometric parameters of D . It particular, inPTheorem 1 we show that D can be explicitly described in the form 2 {f : supj∈J ω∈Ωj m(ω)|f (ω)| < 1 for a suitable partition {Ωj : j ∈ J} of the 1
space Ω and a function m with geometrically determined finite positive upper and lower bounds. The main tools we use in our arguments come from Jordan theory: by Kaup’s celebrated Riemann Mapping Theorem [7], bounded symmetric circular domains can be regarded as unit balls of JB*-triples. In our case this means that the whole function space can be equipped with a ternary product connected naturally with an equivalent norm whose unit ball is D . The first key point is our formula (***) where, lying upon new simple Jordan-algebraic considerations, we identify the extreme points of the dual unit ball of the associated JB*-triple as certain finite linear combinations of point evaluations – a result which generalizes the classical Banach-Stone Theorem (corresponding to the case D = {f : |f | < 1} ). The second key point is the fact that we are able to pass to the second dual whose atoms are closely related to the extrem points of the dual unit ball and the triple product on the original space can be retrieved from the relations between the atoms in the bidual. Concerning the basic facts and notations of JB*-triple theory we refer to the recent survey [10]. 2. Results Definition. Throughout this work let Ω be a locally compact Hausdorff topological space. By a symmetric continuous Reinhardt domain over Ω we mean a bounded symmetric domain D ⊂ C0 (Ω) (the space of all continuous functions Ω → C with {ω ∈ Ω : |f (ω)| ≥ ε} compact ⊂ Ω for any ε > 0 ) equipped with the norm kf k∞ := max |f | ) such that f ∈ D, |g| ≤ f ⇒ g ∈ D
(f, g ∈ C0 (Ω) .
Henceforth we fix a symmetric continuous Reinhardt domain D over Ω . It is wellknown [7] that D is the unit ball for a so-called JB*-norm k.k on C0 (Ω) equivalent to k.k∞ and there exists (a unique) 3-variable operation {...} : C0 (Ω)3 → C0 (Ω) such that (J1) {xyz} = {zyx} , (J2) {(α1 x1 + α2 x2 )(β1 y1 + β2 y2 )(γ1 z1 + γ2 z2 )}=
2 X
αk βℓ γm {xk yℓ zm } ,
k,ℓ,m=1
(J3) {abxyz} = {abxyz} − {xbayz} + {xyabz} , (J4) k{xxx}k = kxk3 , (J5) the spectrum of the operator [x 7→ {aax}] is non-negative for all a, b, x, y, z ∈ C(Ω) and α1 , α2 , β1 , β2 , γ1 , γ2 ∈ C . We shall write briefly E for the JB*-triple C0 (Ω), k.k, {...} . Since the norms k.k, k.k∞ are equivalent on E , by the Riesz Representation Theorem [5], the dual space E ′ consists of all linear functionals of the form [f 7→
Z
f dµ]
(µ ∈ M(Ω)) 2
where M(Ω) denotes the family of all complex-valued regular measures with bounded variation on Ω . That is any element µ ∈ M(Ω) is a σ -additive function Borel(Ω) → C such that µ(B) = supK compact⊂B µ(K) for B ∈ Borel(Ω) := [the smallest σ -ring of subsets of Ω containing Ω and the compact subsets of Ω ]. R We denote the functionalR [E ∋ f 7→ f dµ] simply by dµ , and conveniently we write gdµ := [f 7→ f g dµ] whenever g ∈ B(Ω) where we write B(Ω) for the family of all bounded complex-valued Borel(Ω) -measurableR functions. In terms of measures, the dual norm on E ′ is kdµk := supf ∈Ball1 (E) | f dµ| where Ball1 (E) := {x ∈ E : kxk ≤ 1} . It is well-known that the dual norm of k.k∞ is nothing else as the total variation (which is finite automatically due to the presence of the whole Ω in Borel(Ω) ) nX o kdµk1 = sup |µ(K)| : K finite disjoint family of compact subsets of Ω . K∈K
In order to achieve a parametrized classification of all symmetric continuous Reinhardt domains over Ω , we shall describe the extreme points of Ball1 (E ′ ) in measure theoretical terms. We can regard the space B(Ω) , which contains E , as a subspace of the R bidual E” by identifying the functional [dµ 7→ f dµ] with f ∈ E as usual. In this sense the JB*-triple product {...} extends from E to E” in a separately weak*-continuous manner and E” with the bidual norm and this extended triple product (denoted simply by k.k and {...} , respectively) is a JB*-triple again [3, 2]. According to a celebrated lemma of Friedman-Russo [4, page 79], the atoms of the JB*-triple E” and the extreme points of the unit ball of E ′ are in a remarkable one-to-one correspondence which can be stated as follows. Recall an atom of E” is an element a ∈ E” such that {aaa} = a 6= 0 and {axa} = φa (x)a
(x ∈ E”)
for some (necessarily unique) linear functional φ : E” → C . Then the extremal points of the dual unit ball are given by ext Ball1 (E ′ ) = φa E : a atom ∈ E” .
Any function u ∈ E gives rise to the multiplication operator Mu : f 7→ uf (f ∈ E) . Notice that for |u| ≤ 1 we have |Mu f | ≤ |f | and hence Mu D ⊂ D . That is Mu Ball1 (E) ⊂ Ball1 (E) for max |u| ≤ 1 and, in general, kMu k ≤ max |u|
(u ∈ E) .
Any function g ∈ B(Ω) gives rise to the multipliers Mg′ (dµ) := g dµ
(µ ∈ M(Ω)) ,
M ”g (f ) := [dµ 7→ f (g dµ)
(f ∈ E”) .
We note that Mg′ : E ′ → E ′ , M ”g : E” → E” are linear operators and if u ∈ E then Mu′ , M ”u are the adjoint E ′ → E ′ and biadjoint E” → E” of Mu , 3
respectively. The norm of an adjoint is always the same as the original operator. In particular we have kMu′ k = kM ”u k ≤ max |u| for all u ∈ E . Lemma 1. In general, kMg′ k = supkdµ=1k kg dµk ≤ sup |g| and kM ”g k = supf ∈Ball1 (E”) kMg f k ≤ sup |g| . Proof. Notice that M ”g is the adjoint of Mg′ . Thus kM ”g k = kMg′ k . To show that kMg′ (dµ)k ≤ sup |g|kdµk , we proceed as follows. Fix µ ∈ M(Ω) arbitrarily. It is well-known that dµ = h|d˜ µ| for some function h ∈ B(Ω) with |h| = 1 and µ ˜ ∈ M(Ω)+ := {ν ∈ M(Ω) : ν(B) ≥ 0 (B ∈ Borel(Ω))} . On the other hand, there exists a sequence g1 , g2 , . . . ∈ C0 (Ω) with max |g1 |, max |g2 |, . . . ≤ sup |g| and µ ˜ {ω ∈ Ω : g(ω) 6= limn→∞ gn (ω)} = 0 . Thus Z ′ kMg (dµ)k = kgh d˜ µk = sup | f gh d˜ µ| = f ∈D Z = sup lim | f gn h d˜ µ| = sup lim |dµ(Mg′ n f )| ≤ f ∈D n→∞
f ∈D n→∞
≤ sup lim kdµk kMgn f k ≤ sup lim kdµk max |gn | = f ∈D n→∞
f ∈D n→∞
= max |gn | kdµk . Given any set S ∈ Borel(Ω) , we write 1S for the indicator function of S , that is 1S (ω) := [1 if ω ∈ S, 0 else] (ω ∈ Ω) . By the previous lemma the operator PS := M ”1S is clearly a contractive projection E” → E” since, in general Mg′ 1 Mg′ 2 = Mg′ 1 g2 ,
M ”g1 M ”g2 = M ”g1 g2
we always have PS PT = PS∩T (S, T ∈ Borel(Ω)) . Recall that surjective linear isometries of JB*-triples are automorphisms of the underlying triple product [6]. In particular, because of the above lemma, M ”g is an automorphism of JB*-triple E” whenever g ∈ B(ω) and |g(ω)| = 1 for all ω ∈ Ω , since it is the inverse of the multiplier M ”g . Proposition 1. Suppose a ∈ E” is an atom and S1 , . . . , SN ∈ Borel(Ω) are SN disjoint sets such that . . . , N ) with the k=1 Sk = PΩ and 2Pk a 6= 0 (k = 1, −1 projections Pk := M ”1Sk . Then Pk a is again an k kPk ak = 1 and kPk ak −1 atom in E” with φkPk ak−1 Pk a = kPk ak φa ◦ Pk for any index k . PN itk Proof. For any t1 , . . . , tn ∈ IR the functions gt1 ,...,tN := 1Sk have k=1 e absolute value 1. Hence At1 ,...,tN :=
N X
eitk Pk = M ”gt1 ,...,tN ∈ Aut(E”)
k=1
4
(t1 , . . . , tN ∈ IR)
with the inverse A−1 t1 ,...,tN
= M ”gt1 ,...,tN =
N X
e−itk Pk .
k=1
It follows that the elements at1 ,...,tN := At1 ,...,tN a
(t1 , . . . , tN ∈ IR)
are all atoms of E” since they are automorphic images of the atom a . Thus {at1 ,...,tN xat1 ,...,tN } = φat1 ,...,tN (x)at1 ,...,tN for any x ∈ E” and t1 , . . . , tN ∈ IR . On the other hand, here we have {at1 ,...,tN xat1 ,...,tN } = At1 ,...,tN {a[A−1 t1 ,...,tN x]a} = = φa (A−1 t1 ,...,tN x)At1 ,...,tN a = = φa (A−1 t1 ,...,tN x)at1 ,...,tN . That is, for all real t1 , . . . , tN we have φat1 ,...,tN (x) = φa (A−1 t1 ,...,tN x) =
N X
e−itk φa (Pk x)
k=1
and {
X k
eitk Pk a]x
X
eitℓ Pℓ a]} =
X
e−itk φa (Pk x)
k
ℓ
X
eitℓ Pℓ a .
ℓ
Comparing the coefficients of the above trigonometric polynomials in the second identity, we get {(Pk a)x(Pℓ a)} = {(Pℓ a)x(Pk a)} =
1 1 φa (Pk a)Pℓ a + φa (Pℓ a)Pk a 2 2
for any x ∈ E” and k, ℓ = 1, . . . , N . In particular, for k = ℓ , these relations mean {(Pk a)E”}(Pk a) ⊂ CPk a , that is each element Pk a is some multiple of a suitable atom. Thus the element ak := kPk ak−1 Pk a is an atom in E” . The above equation further implies {(Pk a)(Pk a)(Pk a)} = φa (Pk a)Pk a . Since the number φa (Pk a) belongs to the spectrum of the operator [x 7→ {(Pk a)(Pk a)x}] , which is non-negative by (J5), we have kPk ak3 = k{(Pk a)(Pk a)(Pk a)}k = kφa (Pk a)Pk ak = φa (Pk a)kPk ak . Hence it follows kPk ak = φa (Pk a)1/2 . Thus, with the coefficients λk := φa (Pk a)1/2 > 0 5
(k = 1, . . . , N )
we have a = a0,...,0 =
X
Pk a =
k
X
X
λk ak ,
k
λ2k =
k
X
φa (Pk a) = φa (a) = 1 .
k
It only remains to relate the functionals φak to φa . Again the relation {(Pk a)x(Pk a)} = φa (Pk x)Pk a entails −1 φak (x)ak = {ak xak } = {[λ−1 k Pk a]x[λk Pk a]} = −1 = λ−2 k φa (Pk x)Pk a = λk φa (Pk x)ak
that is φak = λ−1 k φa ◦ Pk (k = 1, . . . , N ) . Corollary 1. The atoms ak := kPk ak−1 Pk a (k = 1, . . . , N ) are pairwise collinear and 1 1 φak (x)aℓ + φaℓ (x)ak (x ∈ E”, k, ℓ = 1, . . . , N ). 2 2 P PN 2 ∈ C with k |ζk | = 1 the element k=1 ζk ak is an atom in E” X ζk φak . φζ1 a1 +···+ζN aN =
{ak xaℓ } = If ζ1 , . . . , ζN and
k
Proof. We have φak = λ−1 k φa ◦ Pk and φek (ak ) = 1 . Since Pk ak = ak and Pk Pℓ = 0 for k 6= ℓ , −1 λ−1 k λℓ φa (Pℓ ak )Pk a + φa (Pk ak )Pℓ a = 2 1 1 (k 6= ℓ) = φak (ak )aℓ + (φaℓ (0) ak = aℓ 2 2
{ak ak aℓ } =
which means the collinearity of the family {ak }N k=1 . Then all the statements follow from the relationship {
X k
X X1 X ζk ζℓ [φak (x)aℓ + φaℓ (x)ak ] = ζk ak x ζℓ aℓ } = ζk ζℓ {ak xaℓ } = 2 ℓ k,ℓ k,ℓ X X ζk φak (x) ζℓ aℓ . = k
ℓ
Since D = Ball1 (E) is a bounded open subset of C0 (Ω) , for some 0 < ε ≤ M < ∞ we have {x : kxk∞ < ε} ⊂ D ⊂ {x : kxk∞ < M } , εkdµk1 ≤ kdµk ≤ M kdµk1
(µ ∈ M(Ω)) .
Lemma 2. If dµ ∈ ext Ball1 (E) and S1 , . . . , SN ∈ Borel(Ω) are pairwise disjoint sets such that 1Sk dµ 6= 0 (k = 1, . . . , N ) then necessarily N ≤ (M/ε)2 . 6
SN S Proof. We may assume k=1 Sk = Ω (by replacing SN with Ω \ k
