A class of archimedean lattice ordered algebras | SpringerLink

Algebra univers. 50 (2003) 305–323 0002-5240/03/040305 – 19 DOI 10.1007/s00012-003-1839-8 c Birkh¨ auser Verlag, Basel, 2003

Algebra Universalis

A class of archimedean lattice ordered algebras Karim Boulabiar and Fatma Hadded Abstract. This paper introduces a ‘new’ class of lattice ordered algebras. A lattice ordered algebra A will be called a pseudo f -algebra if xy = 0 for all x, y in A such that x ∧ y is a nilpotent element in A. Different aspects of archimedean pseudo f -algebras are considered in detail. Mainly their integral representations on spaces of continuous functions, as well as their connection with almost f -algebras and f -algebras. Various characterizations of order bounded multiplicators on pseudo f -algebras are given, where by a multiplicator on a pseudo f -algebra A we mean an operator T on A such that xT (y) = yT (x) for all x, y in A. In this regard, it will be focused on the relationship between multiplicators and orthomorphisms on pseudo f -algebras.

1. Introduction The importance of f -algebras has steadily grown since their introduction in the fifties by G. Birkhoff and R. S. Pierce [5]. Though certain other -algebras seem to be worthy of consideration, such as almost f -algebras, very little attention has been paid to -algebras that are not f -algebras. Very recently, M. Henriksen [11] expressed his wish to see more papers dealing with -algebras rather than f algebras. Besides, he furnished several strong reasons that should motivate workers in -algebras to start thinking beyond the f -algebras context. The present paper intends to contribute to this program by introducing a ‘new’ class of -algebras that need not be f -algebras. The -algebras under consideration in this work are those with the additional property that xy = 0 whenever x ∧ y is a nilpotent element. Such an -algebra will be called a pseudo f -algebra. For the sake of clearness, an example of a pseudo f -algebra which is not an f -algebra is provided next. Let A = C([−1, 1]) be the vector space of all real-valued continuous functions on [−1, 1] and consider the multiplication ∗ defined in A by ⎧ −t ⎪ ⎨ x(s)y(s)ds if −1 ≤ t ≤ 0 (x ∗ y)(t) = 0 ⎪ ⎩ tx(t)y(t) if 0 ≤ t ≤ 1. Presented by M. Henriksen. Received April 22, 2003; accepted in final form July 29, 2003. 2000 Mathematics Subject Classification: 06F25, 47B65, 46J10. Key words and phrases: Almost f -algebra, f -algebra, continuous function, multiplicator, orthomorphism, pseudo f -algebra. 305

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Under ∗, A is a pseudo f -algebra but not an f -algebra. Although the pseudo f -algebra structure seems to be a little far-fetched, we believe that it offers an intersecting side road, in part for clarifying the role of certain technical aspects of the theory of f -algebras, as well as, almost f -algebras. The latter received their name in the sixties from G. Birkhoff [4]. For terminology, notations and properties not explained or proved in this paper, the reader can consult the classical books [1] by C. D. Aliprantis and O. Burkinshaw and [16] by H. Schaefer. 2. Preliminaries A vector lattice (also called a Riesz space) L is said to be archimedean if for each non zero x ∈ L the set {nx : n = ±1, ±2, . . .} has no upper bound in L. All vector lattices and lattice ordered algebras under consideration are real and archimedean. First of all, let us recall the definition of the uniform completion of a vector lattice. Let L be a vector lattice and let Lδ denote the Dedekind completion of L. The closure of L in Lδ with respect to the uniform topology is a uniformly complete vector lattice denoted by Lru . We call Lru the uniform completion of L after J. Quinn in [15]. For the (relatively) uniform topology on vector lattices, see W. A. J. Luxemburg and A. C. Zaanen [12, Section 13]. Next, we discuss (linear) operators on vector lattices. Let L and M be vector lattices with positive cones L+ and M + , respectively, and let T be an operator from L into M . One says that T is order bounded if for each x ∈ L+ there exists y ∈ M + such that |T (z)| ≤ y in M whenever |z| ≤ x in L. The operator T is said to be positive if T (x) ∈ M + for all x ∈ L+ . Every positive operator is of course order bounded. The operator T is called a lattice (or Riesz ) homomorphism if T (x ∧ y) = T (x) ∧ T (y) (equivalently, T (x ∨ y) = T (x) ∨ T (y)) for all x, y ∈ L. The operator T is a lattice homomorphism if and only if T (|x|) = |T (x)| for all x ∈ L. Clearly, every lattice homomorphism is positive and then order bounded. The set Lb (L) of all order bounded operators on L is an ordered vector space with respect to pointwise operations and order. The positive cone of Lb (L) is the subset of all positive operators. An element T in Lb (L) is referred to as an orthomorphism if, for all x, y ∈ L, |T x|∧|y| = 0 whenever |x|∧|y| = 0. Under the ordering and operations inherited from Lb (L), the set Orth(L) of all orthomorphisms on L is an archimedean vector lattice. The absolute value in Orth(L) is given by |T | (x) = |T (x)| for all x ∈ L+ . More about order bounded operators and orthomorphisms on vector lattices can be found in [1] by C. D. Aliprantis and O. Burkinshaw. The following paragraph deals with lattice ordered algebras. The vector lattice A is said to be a lattice ordered algebra (briefly, an -algebra) if there exists an associative multiplication in A with the usual algebra properties such that xy ∈ A+

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for all x, y ∈ A+ . For an -algebra A, we denote N (A) = {x ∈ A : xn = 0 for some n = 1, 2, . . .}, that is, N (A) is the set of all nilpotent elements in A. Also, for a fixed nonnegative integer n, we put Nn (A) = {x ∈ A : xn = 0}. The -algebra A is said to be semiprime (or reduced ) if N (A) = {0}. The algebra A is called an f -algebra if A has the property that x ∧ y = 0 in A implies (xz) ∧ y = (zx) ∧ y = 0 for all z ∈ A+ . It follows that multiplication by an element in an f -algebra is an orthomorphism. If L is a vector lattice then the vector lattice Orth(L) is an f -algebra with respect to the composition as multiplication. Moreover, the identity map IL on L is the multiplicative unit of Orth(L). In particular, the f -algebra Orth(L) is semiprime. In a (not necessarily archimedean) f -algebra A, the equality |xy| = |x| |y| holds for all x, y ∈ A and then squares are positive. The (archimedean) f -algebra A is automatically commutative and satisfies N (A) = N2 (A) = {x ∈ A : xy = 0 for all y ∈ A}. Orthomorphisms and f -algebras are studied extensively in the Ph.D. thesis of de Pagter [14]. An almost f -algebra is an -algebra A such that x ∧ y = 0 in A implies xy = 0. An -algebra is an almost f -algebra if and only if x2 = |x|2 for all x ∈ A. As for f -algebras, any (archimedean) almost f -algebra is commutative and satisfies N2 (A) = {x ∈ A : xy = 0 for all y ∈ A} and N (A) = N3 (A) = {x ∈ A : xyz = 0 for all y, z ∈ A}. For an (almost) f -algebra A, both N (A) and N2 (A) are -ideals, that is, order and ring ideals. For more information about almost f -algebras, the reader is referred to S. J. Bernau and C. B. Huijsmans [3] and K. Boulibiar [6]. At this point, we shall introduce and give the first properties of the class of -algebras that will be surveyed in this paper. Definition 2.1. The -algebra A is said to be a pseudo f -algebra if x ∧ y ∈ N (A) implies xy = 0. It is not hard to see that any (archimedean) pseudo f -algebra is an almost f -algebra. Then any pseudo f -algebra is commutative and has positive squares. However, almost f -algebras need not be pseudo f -algebras as is shown in the next example.

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Example 2.2. Take A = R2 with the coordinatewise vector space operations and partial ordering. Define a multiplication ∗ in A by (a, b) ∗ (a , b ) = (bb , 0) for all (a, b), (a , b ) ∈ A. It is easily checked that A is an archimedean almost f algebra with respect to the multiplication ∗. However, A is not a pseudo f -algebra under ∗ since N (A) = A (that is, x ∧ y ∈ N (A) for all x, y ∈ A) and ∗ is not trivial. It seems natural therefore to ask what is missing for an almost f -algebra to be a pseudo f -algebra. The answer is given in the next theorem. First, let us recall that a positive element z in an -algebra A is called an f -element if (zx)∧y = (xz)∧y = 0 for all x, y ∈ A such that x ∧ y = 0 (see S. Steinberg [19]). Theorem 2.3. Let A be an archimedean -algebra. The following are equivalent. (i) A is a pseudo f -algebra, (ii) A is an almost f -algebra with N (A) = N2 (A), (iii) A is an almost f -algebra such that every positive nilpotent element in A is an f -element. Proof. (i) ⇒ (ii) We already pointed out that any pseudo f -algebra is an almost f -algebra. Obviously, N2 (A) ⊂ N (A). Conversely, let x ∈ N (A). Since x ∧ x = x ∈ N (A), we get x2 = 0 and thus x ∈ N2 (A). (ii) ⇒ (iii) Let 0 ≤ x ∈ N (A) = N2 (A). Hence xy = yx = 0 for all y ∈ A. In particular, x is an f -element in A. (iii) ⇒ (i) Let x, y ∈ A such that x ∧ y ∈ N (A). We claim that xy = 0. Indeed, since N (A) is an order ideal, we get |x ∧ y| ∈ N (A), so that |x ∧ y| is an f -element. It follows that the operator π: A → A z → |x ∧ y| z is an orthomorphism on A. Observe that 2

π 2 (z) = |x ∧ y| z = 0 for all z ∈ A because |x ∧ y| ∈ N (A) = N3 (A). But then π 2 = 0. Consequently, π is a nilpotent element in the semiprime f -algebra Orth(A). We derive that π = 0. Hence, by Proposition 1.13 in [3], we obtain 0 ≤ |xy| = |(x ∧ y)(x ∨ y)| ≤ |(x ∧ y)| |(x ∨ y)| = π(|x ∨ y|) = 0, that is xy = 0. This yields that A is a pseudo f -algebra and we are done.

The equivalence (i) ⇔ (ii) in Theorem 2.3 will be used throughout this paper but sometimes without further mention.

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Finally, by a pseudo f -multiplication ∗ in a vector lattice A, we shall mean a multiplication ∗ under which A is a pseudo f -algebra. In the same way, we define almost f -multiplications and f -multiplications in vector lattices. In the next section, we will investigate pseudo f -multiplications in the space C(Ω) of all realvalued continuous functions on a compact Hausdorff topological space. 3. Pseudo f -multiplications in C(Ω) Throughout this section, C(Ω) stands for the space of all real-valued continuous functions on a compact Hausdorff topological space Ω. The unit function on Ω will be denoted by e. The principal objective of this section is to characterize pseudo f -multiplications in C(Ω). In order to hit this mark, some prerequisites are needed. Let M (Ω) be the topological space of all Radon measures on Ω, that is, M (Ω) is the set of all order bounded functionals on C(Ω) furnished with the weak topology σ(M (Ω), C(Ω)). Equipped with the induced topology, the subset M (Ω)+ of all positive measures in M (Ω) is a topological space. The support of a measure µ in M (Ω)+ is denoted by supp(µ), that is, supp(µ) is the smallest closed subset Ωµ of Ω such that µ(Ω\Ωµ ) = 0. For a subset M = {µt : t ∈ Ω} of M (Ω)+ , we will use the following subsets of Ω UM = {t ∈ Ω : 0 < µt (e)}, SM = {supp(µt ) : t ∈ Ω}, and VM = SM ∩ UM . The Dirac measure supported at t ∈ Ω will be denoted, as usual, by δt . Of course, δt ∈ M (Ω)+ for all t ∈ Ω. In [17], E. Scheffold proved that if ∗ is a non trivial almost f -multiplication in C(Ω), then there exists a continuous function t → µt from Ω into M (Ω)+ , the image of which is denoted by M, such that (x ∗ y)(t) = x(s)y(s)dµt (s) Ω

for all x, y ∈ C(Ω) and t ∈ Ω, and (α) if VM = ∅ then x∗y∗z =0 for all x, y, z ∈ C(Ω), (β) if VM = ∅ then µt = (e ∗ e)(t)δt for all t ∈ VM , so that (x ∗ y)(t) = (e ∗ e)(t)x(t)y(t) for all x, y ∈ C(Ω) and t ∈ VM .

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(see also, [2]). If in particular UM = VM then (x ∗ y)(t) = (e ∗ e)(t)x(t)y(t) for all t ∈ Ω, that is, ∗ is an f -multiplication in C(Ω). One retrieves the well-known representation formula for f -multiplications in C(Ω) due to P. Conrad [10]. Remark also that if the almost f -multiplication ∗ is in addition a pseudo f -multiplication, then VM = ∅. Indeed, otherwise, the condition (α) above implies that all functions in C(Ω) are nilpotent with respect to ∗. Since ∗ is assumed to be a pseudo f multiplication, we get that x ∗ y = 0 for all x, y ∈ C(Ω). In other words ∗ is trivial, which contradicts the hypothesis. We are in position now to prove our first representation theorem of pseudo f multiplications in C(Ω). We indebted ourselves to Professor Egon Scheffold in some steps of the necessary condition. Theorem 3.1. Let ∗ be a non trivial multiplication in C(Ω). Then C(Ω) is a pseudo f -algebra with respect to ∗ if and only if there exists a continuous function t → µt from Ω into M (Ω)+ with image denoted by M, such that (i) (x ∗ y)(t) = Ω x(s)y(s)dµt (s) for all x, y ∈ C(Ω) and t ∈ Ω, (ii) VM = ∅ and SM ⊂ VM , where VM is the closure of VM in Ω, (iii) µt = (e ∗ e)(t)δt for all t ∈ SM . Proof. Assume first that ∗ is a multiplication in C(Ω) such that the conditions (i), (ii) and (iii) hold. Let x, y, z ∈ C(Ω) and t ∈ Ω. Using (i) and (iii), we get that (x ∗ y)(s)z(s)dµt (s) ((x ∗ y) ∗ z)(t) = (x ∗ y)(s)z(s)dµt (s) = Ω SM (e ∗ e)(s)x(s)y(s)z(s)dµt (s) = (x ∗ (y ∗ z))(t). = SM

Therefore the multiplication is associative. By (i), we derive that ∗ is an almost f -multiplication. We claim that ∗ is a pseudo f -multiplication. To this end, we have to prove that if x ∗ x ∗ x = 0 for some x ∈ C(Ω) then x ∗ x = 0. Choose x ∈ C(Ω) so that x ∗ x ∗ x = 0. In view of (ii), VM = ∅ and therefore x(t) = 0 for all t ∈ VM , where we use [17, Korollar 1.6]. Since x is continuous, x(t) = 0 for all t ∈ VM and, by (ii), x(t) = 0 for all t ∈ SM . But then, by (i) x2 (s)dµt (s) = x2 (s)dµt (s) = 0 (x ∗ x)(t) = Ω

SM

for all t ∈ Ω, that is, ∗ is a pseudo f -multiplication, as required. Conversely, assume that ∗ is a pseudo f -multiplication. Since in particular ∗ is a non trivial almost f -multiplication, the remark above together with Scheffold’s theorem proves that there exists a continuous function t → µt from Ω into M (Ω)+

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such that VM = ∅, the condition (i) holds, and µt = (e ∗ e)(t)δt for all t ∈ VM . By continuity, one gets µt = (e ∗ e)(t)δt for all t ∈ VM . Therefore, it remains to show that SM ⊂ VM . Arguing by contradiction, assume that there exists t0 ∈ SM such / VM . By Urysohn’s lemma, one can choose x ∈ C(Ω) so that x(t0 ) = 1 that t0 ∈ and x(t) = 0 for all t ∈ VM . Using once more [17, Korollar 1.6], we get x ∗ x ∗ x = 0 and then x ∗ x = 0 since ∗ is a pseudo f -multiplication. Consequently, x2 (s)dµt (s) = (x ∗ x)(t) = 0 Ω

for all t ∈ Ω. Since x is continuous, the last equality leads to x(t) = 0 for all t ∈ SM . This contradicts the fact that x(t0 ) = 1 and the proof of the theorem is complete. Another representation result of pseudo f -multiplications in C(Ω) in terms of regular elements is given next. First, recall that an element w in a commutative algebra A is said to be regular if aw = 0 for some a ∈ A implies a = 0. Theorem 3.2. Let ∗ be a non trivial multiplication in C(Ω). Then C(Ω) is a pseudo f -algebra with respect to ∗ if and only if there exists a continuous function t → µt from Ω into M (Ω)+ with image denoted by M, such that (i) (x ∗ y)(t) = Ω x(s)y(s)dµt (s) for all x, y ∈ C(Ω) and t ∈ Ω, (ii) VM = ∅ and the restriction of e ∗ e to SM is regular in C(SM ), where SM is the closure of SM in Ω, (iii) µt = (e ∗ e)(t)δt for all t ∈ SM . Proof. Assume that C(Ω) is a pseudo f -algebra with respect to ∗. It follows from Theorem 3.1 that there exists a continuous function t → µt from Ω into M (Ω)+ , the image of which is denoted by M, such that VM = ∅, and conditions (i) and (iii) hold. Let us show that the restriction of e ∗ e to SM is regular in C(SM ). To this end, let x ∈ C(SM ) such that (e ∗ e)(t)x(t) = 0 for all t ∈ SM . By the Tietze-Urysohn extension theorem, x extends to x in C(Ω). Then, it follows from (i) and (iii) that x∗x )(s) x(s)dµt (s) = ( x∗x )(s) x(s)dµt (s) ( x∗x ∗x )(t) = ( Ω SM = (e ∗ e)(s) x(s)3 dµt (s) = (e ∗ e)(s)x(s)3 dµt (s) = 0 SM

SM

for all t ∈ Ω, that is, x ∗x ∗x = 0. But then x ∗x = 0 because ∗ is a pseudo f -multiplication. Therefore, x 2 (s)dµt (s) = ( x∗x )(t) = 0 Ω

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for all t ∈ Ω. By continuity, x 2 = 0 over SM and then over SM . Hence, x(t) = x (t) = 0 for all t ∈ SM . Finally, the restriction of e ∗ e to SM is regular in C(SM ). Conversely, assume that there exists a continuous function t → µt from Ω into M (Ω)+ such that ∗ satisfies the conditions (i), (ii) and (iii). A similar argument to that used in the proof of Theorem 3.1 allows us to conclude that ∗ is associative and then an almost f -multiplication. It remains to show that if x is nilpotent with respect to ∗ then x ∗ x = 0. Obviously, we may assume that x is positive. So, let x ∈ C(Ω)+ such that x ∗ x ∗ x = 0. We claim that x ∗ x = 0. Indeed, observe that 0 = (x ∗ x ∗ x)(t) = (x ∗ x)(s)x(s)dµt (s) Ω (x ∗ x)(s)x(s)dµt (s) = (e ∗ e)(s)x(s)3 dµt (s). = SM

SM

for all t ∈ Ω. We derive that (e ∗ e)(t)x(t) = 0 for all t ∈ SM . By continuity, (e ∗ e)(t)x(t) = 0 for all t ∈ SM . The regularity in C(SM ) of the restriction of e ∗ e to SM implies that x(t) = 0 for all t ∈ SM . But then, for every t ∈ Ω, we have (x ∗ x)(t) = x(s)2 dµt (s) = x(s)2 dµt (s) = 0. SM

Ω

It follows that ∗ is a pseudo f -multiplication and we are done.

4. Theoretical properties of pseudo f -algebras In the previous section, we studied pseudo f -multiplications in spaces of realvalued continuous functions. In the present section, we shall provide some theoretical lattice and algebra properties of pseudo f -algebras. We plunge into the matter with the following proposition. Proposition 4.1. Let A be an archimedean pseudo f -algebra. Then the equivalences (xy)2 = 0 ⇔ xy = 0 ⇔ |x| |y| = 0 hold for all x, y ∈ A. Proof. It is not hard to see that it suffices to establish the arrow (xy)2 = 0 ⇒ |x| |y| = 0. Take x, y in A so that (xy)2 = 0. We have 2

2

0 ≤ (|x| ∧ |y|)4 ≤ |x| |y| = (xy)2 = 0, that is |x| ∧ |y| ∈ N (A). Hence, |x| |y| = 0 and the proposition follows.

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In [20, Theorem 3.4], A. Triki proved that an (almost) f -multiplication in a vector lattice A extends uniquely into an (almost) f -multiplication in the uniform completion Aru of A. Next we prove the corresponding result for pseudo f -multiplications. Corollary 4.2. Let A be an archimedean pseudo f -algebra. Then Aru is furnished with a unique pseudo f -multiplication that extends the multiplication in A. Proof. Since A is in particular an almost f -algebra, the multiplication in A extends uniquely to an almost f -algebra multiplication in Aru . We claim that this extended multiplication is actually a pseudo f -multiplication. To this end, we will prove that N (Aru ) = N2 (Aru ). Fix 0 ≤ x ∈ N (Aru ) and let y, z ∈ A+ such that y ≤ x and z ≤ x. From 0 ≤ (yz)2 ≤ x4 = 0 it follows that (yz)2 = 0. By Proposition 4.1, yz = 0. On the other hand, Lemma 3.2 in [20] yields that yx = sup{yz : z ∈ A+ with z ≤ x}. This shows that yx = 0. Using once more [20, Lemma 3.2], we get x2 = sup{yx : y ∈ A+ with y ≤ x} and then x2 = 0. Accordingly, N (Aru ) = N2 (Aru ). By Theorem 2.3, we derive that the almost f -algebra Aru is a pseudo f -algebra, as required. For an -algebra A we denote A2 = {xy : x, y ∈ A}

and

2 + A+ 2 = {x : x ∈ A }.

Next, we investigate the algebraic and ordered structure of A2 for a uniformly complete pseudo f -algebra A. The corresponding problem for almost f -algebras has been considered in [6] and [7]. Indeed, it is shown in [6, Proposition 3.3] that if A is a uniformly complete almost f -algebra then A2 is a positively generated ordered algebra with A+ 2 as positive cone. However, A2 is not necessarily a vector lattice (see [7, Example 1]). It turns out that if A is a uniformly complete pseudo f -algebra then A2 is a vector lattice. First let us prove a useful lemma. Lemma 4.3. Let A be an almost f -algebra. Then the equalities [(x ∧ y)z]2 = [(xz) ∧ (yz)]2

and

[(x ∨ y)z]2 = [(xz) ∨ (yz)]2

hold for all x, y ∈ A and z ∈ A+ . Proof. It follows from (x − x ∧ y) ∧ (y − x ∧ y) = 0

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that (x − x ∧ y)(y − x ∧ y) = 0. Hence, (xz − (x ∧ y)z)(yz − (x ∧ y)z) = 0. Observe that 0 ≤ xz − (x ∧ y)z

and 0 ≤ yz − (x ∧ y)z.

Therefore, 0 ≤ [(xz ∧ yz) − (x ∧ y)z]2 = [(xz − (x ∧ y)z) ∧ (yz − (x ∧ y)z)]2 ≤ (xz − (x ∧ y)z)(yz − (x ∧ y)z) = 0. We derive that (xz ∧ yz) − (x ∧ y)z ∈ N2 (A). Whence 0 = [(xz ∧ yz) − (x ∧ y)z][(xz ∧ yz) + (x ∧ y)z] = (xz ∧ yz)2 − ((x ∧ y)z)2 , and the first equality of the lemma follows. Replacing in the first equality x and y by −x and −y, respectively, we get the second equality. Proposition 4.4. Let A be a uniformly complete pseudo f -algebra. Then A2 is a semiprime f -algebra with respect to the operations and order inherited from A. Proof. At first, we prove that A2 is a vector lattice. Let x, y ∈ A+ . We claim that 2 2 2 inf{x2 , y 2 } exists in A+ 2 and satisfies inf{x , y } = (x ∧ y) . To this end, observe that x2 ≥ (x ∧ y)2 , y 2 ≥ (x ∧ y)2 and suppose z ∈ A+ so that x2 ≥ z 2 and y 2 ≥ z 2 . Put X = (z − (x ∧ y))+ . Then X = (z − x)+ ∨ (z − y). On the other hand X ≤ z + (x ∧ y) = (x + z) ∧ (y + z). Hence, using Lemma 4.3 X 6 ≤ ((z − x)+ ∨ (z − y))2 (y + z)2 (x + z)2 = (((z − x)+ (y + z)) ∨ (z 2 − y 2 ))2 (x + z)2 .

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Since z 2 − y 2 ≤ 0, we obtain (again by Lemma 4.3) X 6 ≤ ((z − x)+ (y + z))2 (x + z)2 = ((z − x)+ (z + x))2 (y + z)2 = ((z 2 − x2 )+ )2 (y + z)2 = 0. Therefore (z − (x ∧ y))+ = X ∈ N (A) = N2 (A), so that (z − (x ∧ y))+ (z + (x ∧ y)) = 0. Consequently, z 2 − (x ∧ y)2 = (z − (x ∧ y))(z + (x ∧ y)) = −(z − (x ∧ y))− (z + (x ∧ y)) ≤ 0. 2 It follows that (x∧y)2 ≥ z 2 and then inf{x2 , y 2 } exists in A+ 2 and equals to (x∧y) , as required. Since in particular A is an almost f -algebra, A2 is an ordered vector + space with A2 = A+ 2 − A2 . By Proposition 1.1.4 in [13], A2 is a vector lattice with + A2 as positive cone. In other words, A2 is an -algebra. Observe that if xy ∈ A2 such that xy is nilpotent then (xy)2 = 0 and, by Proposition 4.1, xy = 0. Hence A2 is semiprime. Moreover, if x, y ∈ A+ such that inf{x2 , y 2 } = 0 in A2 then x ∧ y is nilpotent in A. Thus xy = 0 and x2 y 2 = 0. This yields that A2 is an almost f -algebra. In summary, A2 is a semiprime almost f -algebra. By [3, Theorem 1.11], A2 is a semiprime f -algebra, as desired.

It should be pointed out that though A2 is a vector lattice on its own, A2 need not be a vector sublattice of A. Indeed, observe that the absolute value in A2 is given by |xy|A2 = |x| |y| for all x, y in A. However, in general, the equality |xy| = |x| |y| fails in pseudo f -algebras (see [6, Example 3.5]). The next lines deal with the multiplication operators in pseudo f -algebras. For an element x in a commutative -algebra A, the multiplication operator πx is defined on A by πx (y) = xy for all y ∈ A. The set of all multiplication operators on A is denoted by M(A), that is, M(A) = {πx : x ∈ A}. It follows from [8, Theorem 1] that if A is an almost f -algebra then M(A) is an f -algebra with respect to the algebra operations and ordering inherited from Lb (A). The absolute value in M(A) is given by |πx | = π|x| for all x ∈ A. In particular, the operator π : A → M(A) x → π(x) = πx

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is a surjective lattice homomorphism. Moreover, π is an algebra homomorphism and the kernel ker(π) of π is precisely N2 (A). In Theorem 2.3, we gave two necessary and sufficient conditions for an almost f -algebra to be a pseudo f -algebra. In the following proposition, we shall present a third one in terms of multiplication operators. Proposition 4.5. Let A be an archimedean almost f -algebra. The following are equivalent. (i) A is a pseudo f -algebra, (ii) the f -algebra M(A) is semiprime. Proof. (i) ⇒ (ii) Assume that A is a pseudo f -algebra and let x ∈ A such that (πx )2 = 0. We claim that πx = 0. Indeed, π(x2 ) = (π(x))2 = (πx )2 = 0, that is, x2 ∈ ker(π) = N2 (A). In other words, x is nilpotent in A. From Theorem 2.3 it follows that x ∈ N2 (A) = ker(π), so that πx = π(x) = 0, as required. Finally, M(A) is semiprime. (ii) ⇒ (i) Let x ∈ N (A) = N3 (A). Hence x3 = 0 and then (πx )3 = π(x)3 = π(x3 ) = 0. Since M(A) is semiprime, we obtain πx = 0. This yields that x ∈ N2 (A) and consequently, N (A) = N2 (A). It follows from Theorem 2.3 that A is a pseudo f -algebra. We end this section by the following proposition. Proposition 4.6. Let A be a uniformly complete pseudo f -algebra. The following are equivalent. (i) the semiprime f -algebra A2 has a multiplicative identity. (ii) the semiprime f -algebra M(A) has a multiplicative identity. Proof. Assume that M(A) has a unit element πe for some e ∈ A and let x, y ∈ A. We have π(e)π(xy) = π(xy), and then y(xe − x) ∈ ker(π) = N2 (A). It follows that (y(xe − x))2 = 0 and, by Proposition 4.1, y(xe − x) = 0. Hence, e2 xy = xy, so that, e2 is the identity element in A2 .

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Conversely, let e ∈ A+ such that e2 is the unit element of A2 . Observe that (πe πx )2 = πe2 x2 = πx2 = (πx )2 for all x ∈ A. Since the f -algebra M(A) is semiprime, πe πx = πx for all x ∈ A+ . Writing x = x+ − x− , we see that πe πx = πx for all x ∈ A. This completes the proof of the proposition. 5. Multiplicators in pseudo f -algebras We call an operator T on a commutative algebra A a multiplicator after E. Scheffold [18] if xT (y) = yT (x) for all x, y ∈ A. Using his representation theorem of almost f -multiplications on C(Ω)-spaces, E. Scheffold proved in [18] that any orthomorphism on a Banach almost f -algebra automatically is an (order bounded) multiplicator. This has been the origin of our interest in order bounded multiplicators on almost f -algebras. However, we will see that the ‘best’ properties of multiplicators hold in pseudo f algebras rather than almost f -algebras. In spite of that, we begin our study with a generalization of Scheffold’s result to an arbitrary almost f -algebra. Proposition 5.1. Let A be an archimedean almost f -algebra. Then any orthomorphism on A is an (order bounded) multiplicator on A. Proof. It is clear that we may prove the result only for positive orthomorphisms. Let 0 ≤ T ∈ Orth(A) and define the bilinear map Φ from A × A into A by Φ(x, y) = xT y for all x, y ∈ A. Obviously, Φ is positive, that is, Φ(x, y) ≥ 0 for all x, y ≥ 0. Besides, if one takes x, y ∈ A such that x ∧ y = 0 then x ∧ T y = 0 and therefore Φ(x, y) = xT y = 0. This implies that Φ is symmetrical (where we use [6, Theorem 2.7] or [9, Corollary 2]). Hence, xT y = Φ(x, y) = Φ(y, x) = yT x for all x, y ∈ A. One derives that T is a multiplicator and the proof of the proposition is complete. The converse of Proposition 5.1 fails even if A is an f -algebra. An example illustrating this fact is given next.

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Example 5.2. Let A = R2 equipped with the coordinatewise operations and ordering. Consider the multiplication ∗ defined in A by (a, b) ∗ (a , b ) = (aa , 0) for all (a, b), (a , b ) ∈ R2 . Clearly, A is an f -algebra with respect to ∗. Let T be the operator on A defined by the 2 × 2 matrix

0 0 T = . 1 0 It an easy task to prove that T is a multiplicator on A. However, T is not an orthomorphism. Observe that the f -algebra A in the example above is not semiprime. Indeed, with the semiprimeness as an extra condition, order bounded multiplicators and orthomorphisms coincide as we shall see next. First, let Multb (A) denote the set of all order bounded multiplicators on a commutative -algebra A. Proposition 5.3. If A is an archimedean semiprime f -algebra then Orth(A) = Multb (A). Proof. The inclusion Orth(A) ⊂ Multb (A) follows from Proposition 5.1 because the f -algebra A is in particular an almost f -algebra. Let us prove the converse inclusion. Take T ∈ Multb (A) and let x, y, z ∈ A such that |x| ∧ |y| = 0. Then |x| ∧ (|z| |T y|) = |x| ∧ |zT y| = |x| ∧ |yT z| = |x| ∧ (|y| |T z|) = 0 It follows that |z| (|x| ∧ |T y|) = 0 for all z ∈ A. Choosing z = |x| ∧ |T y| in the equality above, we get (|x| ∧ |T y|)2 = 0. But then |x| ∧ |T y| = 0 since A is semiprime. Finally, T ∈ Orth(A) and we are done. From Proposition 5.3 above, it follows that Multb (A) is a subalgebra of Lb (A) as soon as A is a semiprime almost f -algebra. However, this result is no longer true if A is an arbitrary almost f -algebra. An example in this direction is given next. Example 5.4. Let A = C([−1, 1]) be the archimedean vector lattice of all realvalued continuous functions on [−1, 1]. Let us define a multiplication ∗ on A as follows ⎧ if t ∈ [−1, 0], ⎪ ⎨0

0 (x ∗ y)(t) = ⎪ x(s)y(s)ds t if t ∈ [0, 1], ⎩ −1

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for all x, y ∈ A. It is easily verified that A is an almost f -algebra with respect to the multiplication ∗. Consider now T, R ∈ Lb (A) defined by 0 x(s)ds (T x)(t) = tx(t) and R(x)(t) = −1

for all x ∈ A and t ∈ [−1, 1]. It is quite simple to prove that T, R ∈ Multb (A). However, T R is given by

0 x(s)ds t (T R)(x)(t) = −1

for all x ∈ A and t ∈ [−1, 1], and it is not a multiplicator. Indeed, 1/4 = (e ∗ T R(i))(1) = (i ∗ T R(e))(1) = 1/3, where i(t) = t and e(t) = 1, for all t ∈ [−1, 1]. Notice that in the example above, the almost f -algebra A is not a pseudo f algebra. Next we will show that if A is a pseudo f -algebra then Multb (A) is closed under composition and then it is a subalgebra of Lb (A). To hit this mark, we need a lemma and a proposition. Lemma 5.5. Let A be an archimedean pseudo f -algebra and x, x , y, y ∈ A. If xy − x y ∈ N (A) (that is, (xy − x y )2 = 0) then xy = x y . Proof. In view of Corollary 4.2, one may assume without restriction that A is uniformly complete. Hence, by Proposition 4.4, A2 is a semiprime f -algebra. But then the equality (xy − x y )2 = 0 implies xy − x y = 0, which gives the desired result. From now on, A is a pseudo f -algebra. Recall that M(A) denotes the semiprime f -algebra of all multiplication operators on A and that the map π defined from A into M(A) by π(x) = πx for all x ∈ A (where πx (y) = xy for all y ∈ A) is a lattice and algebra homomorphism. Proposition 5.6. Let A be an archimedean pseudo f -algebra and T ∈ Lb (A). Then T ∈ Multb (A) if and only if there exists T ∈ Orth(M(A)) such that T π = πT . Proof. Assume that T ∈ Multb (A). Let x ∈ N (A) = ker(π). Then yT (x) = xT (y) = 0 for all y ∈ A. Hence T (x) ∈ N (A) and thus T maps N (A) into N (A). This enables us to define an operator T from M(A) into M(A) by putting T π = πT . We claim that T is an orthomorphism on M(A). To this end, let x, y ∈ A and observe that π(x)T (π(y)) = π(x)π(T y) = π(xT y) = π(yT x) = π(y)T (π(x)).

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Hence, T is a multiplicator on the semiprime f -algebra M(A). Furthermore, T is order bounded. Indeed, choose x ∈ A so that |π(x)| ≤ π(a), for some a ∈ A+ . Hence π((|x| − a)+ ) = 0 and so (|x| − a)+ ∈ N (A). Therefore |x| |T y| − a |T y| = (|x| − a) |T y| = −(|x| − a)− |T y| ≤ 0. for all y ∈ A. Consequently, |π(y)| π(|T x| − |T a|) = |π(y)| π(|T x|) − |π(y)| π(|T a|) = |π(yT x)| − |π(yT a)| = |π(xT y)| − |π(aT y)| = π(|x| |T y|) − π(a |T y|) ≤ 0, so that, (π(|T x| − |T a|))+ |π(y)| = (π(|T x| − |T a|) |π(y)|)+ = 0 for all y ∈ A. In particular, if y = (|T x| − |T a|)+ then

2 π((|T x| − |T a|)+ ) = 0 Since the f -algebra M(A) is semiprime, we obtain [π(|T x| − |T a|)]+ = π((|T x| − |T a|)+ ) = 0 Whence, |T (π(x))| = π |(T x)| ≤ π(|T a|) and T ∈ Lb (M(A)). In summary, T ∈ Multb (M(A)). By Proposition 5.3, we derive that T is an orthomorphism of M(A). Conversely, let T ∈ Orth(M(A)) such that T π = πT . We have to prove that T ∈ Multb (A). Since in particular M(A) is an almost f -algebra, the orthomorphism T is a multiplicator (see Proposition 5.1). So, let x, y ∈ A and observe that π(xT (y) − yT (x)) = π(x)π(T (y)) − π(y)π(T (x)) = π(x)T (π(y)) − π(y)T (π(x)) = 0. This means that xT (y) − yT (x) ∈ ker(π) = N (A), that is, (xT (y) − yT (x))2 = 0. From Lemma 5.5 it follows that xT y = yT x. We derive that T is a multiplicator on A and the proof of the proposition is complete. We are in position now to prove that Multb (A) is an algebra whenever A is a pseudo f -algebra.

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Theorem 5.7. Let A be an archimedean pseudo f -algebra. Then Multb (A) is an ordered subalgebra of Lb (A). Proof. We only have to prove that Multb (A) is closed under composition of operators. Let R, S ∈ Multb (A) and x, y ∈ A. Then xRS(y) = xR(S(y)) = S(y)R(x) = ySR(x)

(1)

and S as defined in the previous proposition. Consider now the orthomorphisms R Since orthomorphisms commute, we get Sπ = S Rπ = πSR. πRS = RπS =R But then π(RS(x) − SR(x)) = 0, so that, RS(x) − SR(x) ∈ ker(π) = N (A). Hence, yRS(x) = ySR(x).

(2)

Combining (1) and (2), we obtain xRS(y) = yRS(x), that is, RS ∈ Multb (A), and we are done.

Notice that Theorem 5.7 does not hold for almost f -algebras that are not pseudo f -algebras (see Example 5.4). The last result of this paper is another characterization of multiplicators on pseudo f -algebras. Theorem 5.8. Let A be an archimedean pseudo f -algebra and T ∈ Lb (A). Then T is a multiplicator if and only if there exists R ∈ Lb (A) such that xT (y) = yR(x) for all x, y ∈ A. Proof. The ‘only if’ part is trivial; the ‘if’ part is much less evident. Let R ∈ Lb (A) such that xT (y) = yR(x) for all x, y ∈ A. Therefore, if x ∈ N (A) = N2 (A) then yT (x) = xR(y) = 0 for all y ∈ A. Hence, T (x) ∈ N (A) for all x ∈ N (A). As in the proof of Proposition 5.6, we can define the operators T from M(A) into M(A) by may be defined from M(A) putting T (π(x)) = π(T (x)) for all x ∈ A. Similarly, R into M(A) by setting R(π(x)) = π(R(x)) for all x ∈ A. Consequently, π(y)T (π(x)) = π(y)π(T (x)) = π(yT (x)) = π(xR(y)) = π(x)π(R(y)) = π(x)R(π(y))

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for all x, y ∈ A. Now, Fix z ∈ A and consider the operator σ defined on M(A) by σ(π(x)) = π(z)T (π(x)) for all x ∈ A. Hence

σ(π(x)) = π(x)R(π(z))

for all x ∈ A. Therefore, σ is an orthomorphism on M(A). It follows from Proposition 5.1 that σ is a multiplicator, that is π(y)σ(π(x)) = π(x)σ(π(y)) for all x, y ∈ A. Replacing σ by π(z)T in the equality above we get π(z) π(y)T (π(x)) − π(x)T (π(y)) = 0 for all x, y ∈ A. Since z is arbitrary in A and M(A) is a semiprime f -algebra, it follows that π(yT (x) − xT (y)) = π(y)T (π(x)) − π(x)T (π(y)) = 0. Consequently, yT (x) − xT (y) ∈ ker(π) = N (A). By Lemma 5.5, yT (x) = xT (y) and T is a multiplicator on A, which is the desired result. Notice that if the pseudo f -algebra A in Theorem 5.8 is semiprime (and then A is an f -algebra) then R = T is the unique operator in Lb (A) such that xT (y) = yR(x) for all x, y ∈ A. At last, we give an example to show that Theorem 5.8 fails to be true in almost f -algebras that are not pseudo f -algebras. Example 5.9. Keep the same almost (but not pseudo) f -algebra A that was previously considered in Example 5.4. Consider the operators T and R defined on A by

0 0 x(s)ds t and R(x)(t) = sx(s)ds T (x)(t) = −1

−1

for all x ∈ A and t ∈ [−1, 1]. It is easily checked that x ∗ T (y) = y ∗ R(x) for all x, y ∈ A. However, T is far from being a multiplicator on A. References [1] C. D. Aliprantis and O. Borkinshaw, Positive operators, Academic Press, Orlando, 1985. [2] M. Basly, Produit naturel pour les F F -alg` ebres de Banach r´ eticul´ ees du type C(X), Func. App. UAM., Varsovia, 18 (1989), 64–66. [3] S. J. Bernau and C. B. Huijsmans, Almost f -algebras and d-algebras, Math. Proc. Comb. Phil. Soc. 107 (1990), 287–308. [4] G. Birkoff, Lattice theory, 3rd. edition, Amer. Math. Soc. Colloq. Publ. No. 25, Providence, Rhode Island, 1967. [5] G. Birkoff and R. S. Pierce, Lattice-ordered rings, An. Acad. Bras Ci., 28 (1956), 41–69.

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[6] K. Boulabiar, A relationship between two almost f -algebra products, Algebra Univ. 43 (2000), 347–367. [7] K. Boulabiar, Products in almost f -algebras, Comm. Math. Univ. Carolinae 41 (2000), 747–759. [8] K. Boulabiar, Positive derivations on Archimedean almost f -rings, Order, 19 (2002), 385–395. [9] G. Buskes and A. van Rooij, Almost f -algebras: commutativity and Cauchy-Schwarz inequality, Positivity 4 (2000), 227–231. [10] P. F. Conrad, The additive group of an f -ring, Canad. J. Math. 15 (1974), 1157–1168. [11] M. Henriksen, Old and new unsolved problems in lattice-ordered rings, J. Martinez (ed.), Ordered Algebraic Structures, Kluwer (2000), 11–17. [12] W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971. [13] P. Meyer-Nieberg, Banach Lattices, Springer Verlag, Berlin, Heidelberg, New York, 1991. [14] B. de Pagter, f -Algebras and orthomorphisms, Thesis, Leiden, 1981. [15] J. Quinn, Intermediate Riesz spaces, Pacific J. Math. 56 (1975), 255–263. [16] H. H. Schaefer, Banach lattices and positive operators, Springer Verlag, Berlin Heidelberg New York, 1974. [17] E. Scheffold, F F -Banachverbandsalgebren, Math. Z. 177 (1981), 193–205. ¨ [18] E. Scheffold, Uber den ordnungsstetigen bidual von F F -Banachverbandsalgebren, Preprint Nr.1418, Technische Hochschule Darmstadt, 1991. [19] Steinberg, S., On the unitability of a class of partially ordered rings that have squares positive, J. Algebra 100 (1986), 325–343. [20] A. Triki, On algebra homomorphisms in complex almost f -algebras, Comm. Math. Univ. Carolinae 43 (2002), 23–31. [21] A. C. Zaanen, Riesz spaces II, North-Holland, Amesterdam, 1983. Karim Boulabiar Institut Pr´ eparatoire aux Etudes Scientifiques et Techniques, Université de Carthage, PB 51, 2070-La Marsa, Tunisia e-mail : [email protected] Fatma Hadded Institut Supérieur des Sciences Appliquées et de Technologie, Université du Centre, Cité Taffala, 4003-Sousse Ibn Khaldoun, Sousse, Tunisia e-mail : [email protected]

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