On the structure of almost " algebras

On the structure of almost f-algebras KARIM BOULABIAR and ELMILOUD CHIL Abstract. This paper is mainly concerned with a representation theorem for almost f-algebras.

1. Introduction The theory of almost f -algebras, introduced by Birkho¤ in 1967 [5], has been, in our opinion, really launched in 1981 thanks to the fundamental paper of Sche¤old [13], in which the author established the following representation formulas: let X be a compact Hausdor¤ space and C (X) be the Banach algebra of real continuous functions on X and assume that C (X) is furnished with an almost f -algebra multiplication : Then there exist a family of positive measures ( x : x 2 X) and a positive function 2 C (X) such that Z (f g) (x) = f (s)g(s)d x (s) X

and

(f

g h) (x) =

Z

(s) f (s)g(s)h(s)d

x (s)

X

hold for all x 2 X and f; g 2 C (X) : Recently in the important article [8], Buskes and van Rooij generalized Sche¤old’s result to the more general structure of almost f -algebras as follows: let A be an Archimedean almost f -algebra with respect to a multiplication : Then there exist (i) an extremally disconnected compact Hausdor¤ (or Stonean) space X; (ii) a lattice homomorphism from A onto an order dense vector sublattice of C 1 (X) (the Banach algebra of all continuous functions ' : X ! [ 1; +1] for which the open subset fx : ' (x) 6= 1g is dense in X); (iii) a positive operator T from the linear 2000 Mathematic Subject Classi…cation: 06F25, 46A40. 1

2

KARIM BOULABIAR AND ELMILOUD CHIL

span of f (f ) (g) : f; g 2 Ag in C 1 (X) into A; and (iv) 0 such that f

2 C 1 (X) ;

g = T ( (f ) (g))

and f g h = T ( (f ) (g) (h)) hold for all f; g; h 2 A: The proofs of the aforementioned results both rely on the representation theory (i.e. they use theorems to the e¤ect that the vector lattice under investigation has an “isomorphic”copy, the elements of which are real continuous functions). Our principal purpose in this paper is to establish an abstracted version of Buskes and van Rooij theorem. In contrast to the original approach of Buskes and van Rooij, our approach is purely order theoretical and algebraic in nature and doesn’t involve any representation theorems. 2. Preliminaries Throughout this paper, we use the unavoidable work [1] as a starting point and we refer the reader to this monograph for unexplained terminology and notations about vector lattices and order bounded operators. All vector lattices and lattice-ordered algebras under consideration are supposed to be Archimedean. Let us recall some of the relevant notions. A (real) vector lattice (or Riesz space) A which is simultaneously an associative algebra with the property that f g 0 for all 0 f; g 2 A (equivalently, jf gj jf j jgj for all f; g 2 A) is called a lattice-ordered algebra (Riesz algebra, brie‡y `-algebra). The `-algebra A is said to be an f -algebra whenever f ^ g = 0 and 0 h 2 A imply f h ^ g = hf ^ g = 0: An almost f -algebra A is an `-algebra with the additional property that f ^ g = 0 in A implies f g = 0 (equivalently, jf j2 = f 2 for all f 2 A). An `-algebra A is called a d -algebra whenever f ^ g = 0 and 0 h 2 A imply f h ^ gh = hf ^ hg = 0 (equivalently, jf gj = jf j jgj for all f; g 2 A). Any f -algebra is an almost f -algebra and d-algebra, but not conversely. An almost f -algebra need not be a d-algebra and vis versa. Both f -algebras and almost f -algebras are automatically commutative and have positive squares but these properties fail in d-algebras. A d-algebra which is commutative or has positive squares is an almost f -algebra. More about almost f -algebras and d-algebras can be found in [2,9] and we refer to [11] for basic concepts concerning f -algebras. In the next lines, we recall the de…nition and some elementary properties of orthomorphisms. Our main reference is [11]. An orthomorphism of the

ON THE STRUCTURE OF ALMOST F-ALGEBRAS

3

vector lattice A is an order bounded operator on A with the property that jf j ^ jgj = 0 implies j (f )j ^ jgj = 0: Every positive orthomorphism is a lattice homomorphism, meaning that if f ^ g = 0 then (f ) ^ (g) = 0 (equivalently, j (f )j = (jf j) for all f 2 A). The collection Orth(A) of all orthomorphisms on A is a vector lattice with respect to the order inherited from Lb (A) (the ordered vector space of all order bounded operators on A) with the following supremum and in…mum (

1

_

2 ) (f )

=

1

(f ) _

2

(f )

;

(

1

^

2 ) (f )

=

1

(f ) ^

2

(f )

for all 1 ; 2 2 Orth(A) and 0 f 2 A: Moreover, Orth(A) is an f -algebra with respect to the composition and the identity map IA of A is a unit element of Orth(A): Every orthomorphism of A is order continuous (this means that inf (j (f )j) = 0 for every downward directed net 0 f 2 A such that f # 0). Therefore if

1;

2

2 Orth(A) such that

1

=

2

holds

on an order dense vector sublattice of A then 1 = 2 (we say that a vector sublattice L of A is order dense in A whenever for every 0 < f 2 A there exists g 2 L satisfying 0 < g f or, equivalently, there exists an upward directed net 0 g 2 L such that g " f ). If A is supposed to

be, in addition, an f -algebra then for every f 2 A; the map f de…ned by f (g) = f g for all g 2 A is an orthomorphism of A: Furthermore, if A is an f -algebra with unit element then the mapping f ! f from A into Orth(A) is a lattice and algebra isomorphism. In particular, for every 2 Orth(A) there exists a unique element f 2 A such that = f and is a positive orthomorphism if and only if f 0: Let us recall some facts about the universal completion of a vector lattice. A Dedekind complete vector lattice is called universally complete whenever every set of pairwise disjoint positive elements has a supremum. Every vector lattice A has a universal completion Au ; meaning that there exists a unique (up to a lattice homomorphism) universally complete (and therefore Dedekind complete) vector lattice Au so that A can be identi…ed with an order dense vector sublattice of Au : Moreover, Au is furnished with a canonical multiplication, under which Au is an f -algebra with unit element. Since we wish to avoid representation in this paper, we adhere to the representation-free approach to the existence of universal completion presented in [1, § 8, Exercise 13]. At this point, let A be the Dedekind completion of the vector lattice A: The closure of A in A with respect to the (relatively) uniform topology is a uniformly complete vector lattice denoted by Aru (for de…nition and basic properties of the (relatively) uniform topology we refer to [10, § 16]). According to [12, De…nition 2.12], Aru is the uniform completion of A: It is

4


easily veri…ed that (Aru ) = A and (Aru )u = Au : On the other hand, it is well-known that if A is an `-algebra then the multiplication in A extends uniquely to a multiplication in Aru in such a fashion that Aru is again a (uniformly complete) `-algebra. Moreover, if A is an f -algebra (respectively, almost f -algebra, d-algebra) then so is Aru (for details see [14, Theorem 3.4]). For these reasons and for the sake of notation’s simpleness, we can assume, without loss of generality, that all vector lattices and `-algebras considered in this work are uniformly complete. Now, let A be an f -algebra with respect to a multiplication : In the next theorem, we will establish a relationship between the product f g of two elements f; g 2 A and their product f g in the f -algebra Au ; universal completion of A: Theorem 1. Let A be an f -algebra with respect to a multiplication : Then there exists a positive element in the f -algebra Au ; universal completion of A; such that f g = fg for all f; g 2 A: Proof. It is well-known that the multiplication in A extends uniquely to an f -algebra multiplication in the Dedekind completion A of A in such a manner that A becomes a subalgebra of A (for details see [4,11]). Furthermore, since A is an order dense vector sublattice of Au and Au is Dedekind complete, A is also an order dense vector sublattice of Au : Actually, A is the order ideal generated by A in Au : Therefore, the universal completion of A is precisely Au : Consequently, we can assume in the remainder of the proof that A is Dedekind complete. At this point, for each f 2 A; we denote by f the orthomorphism of A de…ned by f (g) = f g for all g 2 A: In view of [1, Theorem 8.28], f extends uniquely to an orthomorphism of Au ; denoted again by f : As Au is an f -algebra with unit element, there exists a unique (f ) 2 Au such that (f ) g for all g 2 Au : Hence, a mapping is automatically de…ned f (g) = by putting : A ! Au f 7! (f ) : Obviously, is a positive operator. Let f 2 A and g 2 Au such that f ^ g = 0 in Au : We claim that (2.1)

(f ) ^ g = 0

u

in A and therefore is, in particular, a lattice homomorphism. To show this, de…ne g 2 Orth (Au ) by putting g (h) = gh for all h 2 Au : So (2.2)

f^

g

(h) = f ^ gh = 0


h 2 Au : Now, let 0

holds for all 0

h

(where we use that f

^

g

h

h 2 A: It follows from (2.2) that

(f ) ^

g

(h) = 0

2 Orth (Au )). Whence

(h) =

f

(h) ^

= (h f ) ^

5

g

h) ^

(h) = (f

g

(h) =

h

(f ) ^

g

(h)

g

(h) = 0

holds in Au : In other words, f ^ g = 0 on the order dense vector sublattice A of Au : Since f ^ g 2 Orth (Au ) ; we get f ^ g = 0: Finally, if we denote the unit element of Au by e then (f ) ^ g = (f ) e ^ ge =

f

(e) ^

g

(e) =

f

^

g

(e) = 0

as required. Observe now that the condition (2.1) implies that is order continuous. To see this, it su¢ ces to reproduce the proof of the order continuity of orthomorphisms (see [1, Theorem 8.10]). In summary, is an order continuous lattice homomorphism. So, according to [1, Theorem 7.20], the formula (g) = sup f (f ) : f 2 A and 0

f

gg ;

g 2 Au

de…nes an extension of from Au into itself so that this extended operator is again an order continuous lattice homomorphism. Furthermore, from (2.1), it follows immediately that this extension is also a positive orthomorphism of Au : But Au is an f -algebra with unit element. Consequently, there exists 0 2 Au such that (f ) = f for all f 2 Au : Finally, if f; g 2 A then f

g=

f

(g) = (f ) g = f g

and we are done. Note that Theorem 1 shows, in particular, that the multiplication in A extends uniquely to an f -algebra multiplication in the universal completion Au of A so that A is a subalgebra of Au : The next paragraph deals with the set of multiplicative operators in an almost f -algebra A; whose multiplication will be denoted by : For every f 2 A; we associate (f ) the element of Lb (A) de…ned by (f ) (g) = f

g

for all g 2 A: Observe that (f ) = (f + ) (f ) and therefore (f ) is a positive operator as soon as 0 f 2 A: We obtain immediately that the map : A ! Lb (A) f 7! (f ) is a positive operator, the range of which will be denoted by (A) : Putting T = IA in [6, Theorem 4.4], we obtain the following theorem.

6


Theorem 2. Let A be an almost f -algebra. Then (A) is an f -algebra with respect to the order inherited from Lb (A) and the composition as multiplication. Moreover, is a lattice and algebra homomorphism from A onto (A) : Recall that A is assumed to be uniformly complete. Therefore (A) is also uniformly complete. This follows from the fact is a lattice homomorphism (see [10, Theorem 59.3]). Combining Theorem 1 and Theorem 2, we obtain straightforwardly the next corollary, which turns out to be useful for later purposes. Corollary 1. Let A be an almost f -algebra with respect to a multiplication : Then there exists a positive element in the f -algebra (A)u ; universal completion of (A) ; such that (f

g) =

(f )

for all f; g 2 A (the multiplication in

(g) =

(f ) (g)

u

(A) is denoted by juxtaposition):

Finally, we state a proposition, which is important in the context of our problem. For the proof, see [7, Proposition 3]. Proposition 1. Let A be a vector sublattice of an f -algebra B with unit element. Then the set Ap = ff g : f; g 2 Ag is a vector sublattice of B with As = ff 2 : f 2 Ag as positive cone. 3. A representation theorem for almost f -algebras Let A be a (uniformly complete) almost f -algebra with respect to a multiplication : For every f 2 A; we associate (f ) the element of Lb (A) de…ned by (f ) (g) = f g for all g 2 A: It is shown in Theorem 2 that the range of the operator

(A) = f (f ) : f 2 Ag : A ! Lb (A) f 7! (f )

is a uniformly complete f -algebra with respect to the order inherited from Lb (A) and the composition as multiplication. Furthermore, again by Theorem 2, is a lattice and algebra homomorphism from A onto (A) : At this point, for each f; g 2 A; we put S (f; g) = supff cos + g sin : 0

2 g:

This supremum exists in A because A is uniformly complete. To be more precise, S (f; g) is the uniform limit of the sequence 2k 2k Sn (f; g) = supff cos n + g sin n : k = 0; 1; ::; 2n g; n 2 f0; 1; ::g 2 2


7

(for details we refer to [3, § 2]). Since (A) is also uniformly complete, S ( (f ) ; (g)) exists in (A) : Furthermore (3.1)

S ( (f ) ; (g)) =

(S (f; g)) :

Indeed, we have (Sn (f; g)) = Sn ( (f ) ; (g)) ;

n 2 f0; 1; ::g:

because is a lattice homomorphism. On the one hand, it is easily checked that the sequence (Sn (f; g)) converges uniformly to (S (f; g)) : Moreover, Sn ( (f ) ; (g)) converges uniformly to S ( (f ) ; (g)) : The formula (3.1) follows from uniqueness of uniform limits. Consider now the universal completion (A)u of (A) : We recall that (A)u is an f -algebra with unit element, multiplication of which will be denoted by juxtaposition. It follows immediately from Proposition 1 that the set (A)p = f (f ) (g) : f; g 2 Ag : is a vector sublattice of

(A)u with (A)s =

(f )2 : f 2 A

as positive cone. We have gathered now all ingredients for the main result of this paper. Theorem 3. Let A be a vector lattice, equipped with a multiplication : Then A is an almost f -algebra with respect to if and only if there exist (i) a universally complete f -algebra B with multiplication denoted by juxtaposition; (ii) a lattice homomorphism from A into of B such that B is the universal completion of (A) ; (iii) a positive operator T from (A)p into A; and (iv) a positive element of B, such that (3.2)

f

g = T ( (f ) (g))

and (3.3)

(f

g) h = T (

(f ) (g) (h))

hold for all f; g; h 2 A: Proof. The “if”part is almost obvious. The “only if”part needs proof. So, assume that A is an almost f -algebra with respect to the multiplication and let f; g 2 A such that (f )2 = (g)2 in B = (A)u : Since B has a unit element and therefore semiprime, we get j (f )j = j (g)j : But is a lattice

8


homomorphism, whence and so f

f

(jf j) =

g g = jf j jf j =

(jf j

(jgj) : This shows that jf j jgj jgj = (jf j

jgj) (jf j + jgj) = 0:

jgj 2 ker

jgj) (jf j + jgj)

We infer that f f = g g: Consequently, we can de…ne a map T from the positive cone (A)s of (A)p into A+ by putting T (f )2 = f f: On the other hand, if f; g 2 A then, in view of [3, Theorem 5.2], we have (f )2 + (g)2 = S ( (f ) ; (g))2

(3.4)

It follows from (3.1) and (3.4) that (f )2 + (g)2 =

(3.5)

(S (f; g))2 :

Moreover, it shown by the …rst author in [6, Theorem 3.1] that (3.6)

f

f + g g = S (f; g) S (f; g) :

Therefore, combining (3.5) and (3.6), we get T

(f )2 + (g)2 = T

(S (f; g))2 = S (f; g) S (f; g) = f

f + g g:

Consequently, T is additive and so it extends uniquely to a positive operator from (A)p into A [1, Theorem 1.7]. This extension is denoted again by T and satis…es ! 2 2 (f ) + (g) (f ) (g) T ( (f ) (g)) = T 2 2 f +g 2

=

f +g 2

f

g 2

f

g 2

for all f; g 2 A: The proof of (3.2) is now complete. At this point, it is shown in Corollary 1 that there exists 0 (A)u such that (f

g) =

(f )

(g) =

=f

g

2B=

(f ) (g)

for all f; g 2 A: Consequently, (f

g) h = T ( (f = T (( (f ) =T(

g) (h)) (g)) (h))

(f ) (g) (h))

holds for all f; g; h 2 A and the proof of the theorem is now complete. Since any commutative d-algebra is an almost f -algebra, we obtain a similar result for commutative d-algebras. The details follow.


9

Corollary 2. Let A be a vector lattice, furnished with a multiplication : Then A is a commutative d-algebra with respect to the multiplication if and only if there exist (i) a universally complete f -algebra B with unit element, multiplication of which is denoted by juxtaposition; (ii) a lattice homomorphism from A into B such that B is the universal completion of (A) ; (iii) a lattice homomorphism T from (A)p into A; and (iv) a positive element of B; such that f

g = T ( (f ) (g))

and (f

g) h = T (

(f ) (g) (h))

hold for all f; g; h 2 A: Proof. Assume that A is a commutative d-algebra with respect to the multiplication . Since any commutative d-algebra is an almost f -algebra, we can apply Theorem 3 to A: So, take B; ; T and as previously de…ned in the Theorem 3. We claim that T is a lattice homomorphism. To this end, let f; g 2 A: So jT ( (f ) (g))j = jf gj and, as A is a d-algebra, we obtain

jT ( (f ) (g))j = jf j jgj = T ( (jf j) (jgj))

= T (j (f )j j (g)j) = T (j (f ) (g)j)

as required. The converse of the corollary is obvious. 4. The Dedekind completion of almost f-algebras In the survey paper [9] it was conjectured by Huijsmans that if A is an almost f -algebra (respectively, a d-algebra) then there exists a multiplication in the Dedekind completion A of A; extending the multiplication in A; such that A is an almost f -algebra (respectively, a d-algebra) with respect to this extended multiplication. This conjecture has been proven by Buskes and van Rooij in [8, Theorem 4.1] for the class of almost f -algebras. However, their proof makes use of representation theory which, as previously mentioned in the introduction , we wish to avoid in this work. For this reason, we reproduce this proof using purely algebraic and order theoretical means. We proceed to the details. Theorem 4. Let A be an almost f-algebra. Then the multiplication in A extends to an almost f-algebra multiplication in A :

10


Proof. Consider B; ; T and as previously de…ned in the Theorem 3. In view of [1, Theorem 7.17], extends to a lattice homomorphism e from A into B: Moreover, from the fact that A is majorizing in A ; it follows that (A) is majorizing in e A : Therefore, (A)p is majorizing in e A p : According to the Kantoroviµc theorem [1, Theorem 2.8], T extends to a positive operator Te from e A into A : De…ne now a multiplication p

in A by putting

(4.1)

f

g = Te (e (f ) e (g))

for all f; g 2 A :

We assert that A is an almost f -algebra with respect to the multiplication de…ned by (4.1). The di¢ cult part of the proof is to establish that is associative. To do this, let 0 f; g 2 A : As (A) is order dense in B and therefore in e A ; (A)p is order dense in e A p : Hence there exists 0 (f ) (g ) 2 (A)p an upward directed net such that (f ) (g ) " e (f ) e (g) for all

and

holds for all : On the other hand, it is easy to see that e Te extends to e A p : Consequently, by Corollary 1 and (3.2)

T

e (f ) e (g) in e A therefore

p

: In particular,

(f ) (g )

e Te ( (f ) (g ))

(f ) (g ) =

(f )

=(

(g ) =

e Te (e (f ) e (g))

(f

T ) ( (f ) (g ))

g )=

(T ( (f ) (g )))

e Te (e (f ) e (g))

holds for all : According to the order continuity of the f -algebra multiplication in B; we get (4.2)

e (f ) e (g)

e Te (e (f ) e (g)) :

Conversely, since (A)p is majorizing in e A p ; there exists 0 ; 2A such that e (f ) e (g) ( ) ( ) and so 0 ( ) ( ) e (f ) e (g) 2 e A p : On the other hand, it follows from Proposition 1 that e A p is a vector sublattice of B with e A s as positive cone. Therefore, there exists 0 h 2 A such that ( ) ( ) e (f ) e (g) = e (h)2 : Hence, by (4.2), we get e (h)2

e Te

e (h)2


11

which implies that e (f ) e (g)

( ) ( )

=( =

e Te ( ( ) ( ) T ) ( ( ) ( ))

( ) ( )

Whence e Te (e (f ) e (g))

(4.3)

e (f ) e (g))

e Te (e (f ) e (g))

e Te (e (f ) e (g)) :

e (f ) e (g) :

Combining (4.2) with (4.3) we deduce that

e Te (e (f ) e (g)) = e (f ) e (g)

(4.4)

holds for all f; g 2 A : Finally, let f; g; h 2 A : From (4.1) it follows that (f

g) h = Te (e (f

g) e (h))

= Te e Te (e (f ) e (g)) e (h)

= Te

Hence, using (4.4) (f and the theorem follows.

e Te (e (f ) e (g)) e (h)

g) h = Te ( e (f ) e (g) e (h))

So far we are unable to prove or disprove the corresponding result for the class of d-algebras, the main reason being that a d-algebra need not be commutative. This means that the method used for almost f -algebras does not work in the case of d-algebras. Nevertheless, if we impose this additional condition then the situation improves as it is shown in the following corollary. Corollary 3. Let A be a commutative d-algebra. Then the multiplication in A extends to a d-algebra multiplication in A : Proof. Keep B; ; T and as previously de…ned in the Corollary 2. Since any commutative d-algebra is an almost f -algebra, we deduce from the proof of Theorem 4 that extends to a lattice homomorphism e from A into B and T extends to a positive operator Te from e A p into A such that f g = Te (e (f ) e (g)) and (f g) h = Te ( e (f ) e (g) e (h))

12


hold for all f; g; h 2 A : On the other hand, T is a lattice homomorphism. Therefore, according to [1, Theorem 7.17], Te can be chosen to be a lattice homomorphism. The conclusion follows from the converse of Corollary 2 and the proof is complete. (1) (2) (3) (4)

(5) (6) (7) (8) (9)

(10) (11) (12) (13) (14)

References Aliprantis, C.D. and O. Borkinshaw, Positive operators, Academic Press, Orlando, 1985. Bernau, S.J. and C.B. Huijsmans, Almost f-algebras and d-algebras, Math. Proc. Camb. Phil. Soc. 107 (1990), 287-308. Beukers, F., C.B. Huijsmans and B. de Pagter, Unital embedding and complexi…cation of f-algebras, Math, Z., 183 (1983), 131-144. Bigard, A., K. Keimel and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer-Verlag, BerlinHeidelberg-New York, 1977. Birkho¤, G., Lattice Theory, 3rd. edition Am. Math. Soc. Colloq. Publ. No. 25, Providence, Rhode Island, 1967. Boulabiar, K., A relationship between two almost f-algebra products, Algebra Univ., 43 (2000), 347-367. Boulabiar, K., On the positive orthosymmetric bilinear maps, Submitted. Buskes, G. and A. van Rooij, Almost f-algebras: structure and the Dedekind completion, Positivity, 4(3) (2000), 233-243. Huijsmans, C. B., Lattice ordered algebras and f-algebras: A survey, in: Studies in Economic Theory 2, Positive Operators, Riesz Spaces and Economics, Springer, Berlin (1991). Luxemburg, W.A.J. and A.C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971. de Pagter, B., f-algebras and orthomorphisms, Thesis, Leiden, 1981. Quinn, J., Intermediate Riesz spaces, Paci…c J. Math. 56 (1975), 225-263. Sche¤old, E., FF-Banachverbandsalgebren, Math. Z., 177 (1981), 193-205. Triki, A., On algebra homomorphisms in complex f–algebras, Preprint, University of Tunis, (1998).

Département des Classes Préparatoires, Institut Préparatoire aux Etudes Scientifiques et Techniques, BP 51, 2070-La Marsa, Tunisia E-mail address: [email protected] Département de Mathématiques, Institut Préparatoire aux Etudes d’Ingénieur de Tunis, 2 rue Jawaher Lel Nehro, 1008-Monflery, Tunisia E-mail address: [email protected]