Characterizations of finite dimensional Archimedean vector lattices

RC

...

|{z} n limes

and let N = {1, 2,...}. For a set A, An denotes the cartesian product A x ... x A, n G N.

The Gelfand-Mazur Theorem states that if A is an associative normed real division algebra, then A is isomorphic to R, C or the quaternion field. For the details, we refer to [2]. For the case of lattice-ordered algebras, Huijsmans [3] proved that an Archimedean lattice-ordered algebra with unit element e > 0 in which every positive element has a positive inverse is lattice and algebra isomorphic to R. Uvar [4] gave an alternative proof to the result of Huijsmans for Banach lattice algebras. Later on, these two results are generalized and combined in [5] by using an easy observation as follows:

Theorem 1. Let A be an Archimedean lattice-ordered algebra with unit element e >  0. Then the following statements are equivalent:

• (i)    Every positive element has a. positive inverse.

• (ii)    A is a d-algebra and every positive element has a positive inverse.

• (iii)    A is an f-algebra and each positive element has a positive inverse.

• (iv)    A is an almost f -algebra and each positive element has an inverse.

• (v)    A is an almost f-algebra and each nonzero element has an inverse.

• (vi)    A is order and algelera isomorphic to R.

• 2.    Preliminaries

If one of the statements above is satisfied and A is a normed lattice-ordered algebra with kek = 1. ihen A is also iso metric to R.

As far as we know, no attention at all has been paid in the literature to the problem when a lattice-ordered algebra is of finite dimension. The aim of this paper is to give a positive answer to this problem. In connection with our problem, Bresar studied the class of finite dimensional spaces which are zero product determined. Recall that an algebra A over a field K is said to be zero product determined if for every bilinear map f : A x A ^ B, where B is an arbitrary vector space over K. with the propcTty that for all x,y E A. f (x,y) = 0 wheilever xy = 0, is of the form f (x,y) = Ф(ху) for some linear map Ф : A ^ B. This concept was introduced in [1]. The original motivation for this concept was problems on the zero product preserving linear maps. Recently, Bresar [1] proved the following result.

Theorem 2. A finite dimensional algebra is zero product determined if and only if it is generated by idempotents.

Bresar pointed out that the main purpose of the paper [1] was to find new examples of zero product determined algebras, but ultimately it is restricted to an unexpected characterization of finite-dimensional algebras that are generated by idempotents and the initial problem of finding new examples still remains open. Moreover, the problem of finding other classes of algebras for which the characterization of Theorem 1 holds is fully open.

As an application of our study, we will give a new class of non-finite dimensional zero product algebras for which the characterization of the previous theorem holds.

In order to avoid unnecessary repetitions, we assume that all vector lattices under consideration are Archimedean.

In the following lines, we recall some definitions and basic facts about vector lattices, lattice-ordered algebras and multilinear maps.

For the unexplained terminology on vector lattices, lattice-ordered algebras and multilinear maps, we refer the reader to [6, 7, 8, 9].

Given a vector lattice A. the set A+ = {a E A : a > 0} is called the positiue cone of A. Let a,e E A+. then e is called a. component of a if e Л (a - e) = 0.

An algebra A which is simultaneously a vector lattice such that the partial ordering and the multiplication in A are compatible. that is a,b E A+ im_rflies ab E A+. is called lattice-ordered algebra (briefly a. '-algebra}. Ari '-algebra. A is called an f-algebra if A verifies the property that a Л b = 0 and c >  0 imply ac Л b = ca Л b = 0. Any /-algebra is automatically commutative and has positive squares. An '-algebra A is called an almost f-algebra whenever it follows from a Л b = 0 th at ab = ba = 0. A ri '-algebra. A is called a d-algebra if A verifies the property that a Л b = 0 arid c > 0 imiAv ac Л bc = ca Л cb = 0.

The vector lattice A is called Dedekind complete if for each non-empty subset B of A which is bounded above. sup B exists in A. The vector lattice A is called laterally complete if every orthogonal system in A has a supremum in A. If A is Dedekind complete and laterally complete, then A is called universally complete. Every vector lattice A has a universal completion Au, this means that there exists a unique (up to a lattice isomorphism) universally complete vector lattice Au such that A can be identified with an order dense sublattice of Au (see [6, Section 8, Exercise 13] for an interesting approach to the existence of the universal completion by using orthomorphisms).

A vector lattice A is said to have the countable sup property, if whenever an arbitrary subset D has a supremum, then there exists an at. most, countable subset C of D with sup C = sup D. A Dedekind complete vector lattice with the countable sup property is called super Dedekind complete vector lattice.

Let A be a vector lattice. A subset S of A+ is called an orthogonal system of A if 0 / S and u Л v = 0 for each pair (u, v) of distinct elements in S. It follows from Zorn’s lemma that every orthogonal system of A is contained in a. maximal orthogonal system.

A subset S in a vector lattice E is called solid, if it follows from |u| 6 |v| iri E and v E S that u E S. A solid vector subspace of a vector lattice is called an ideal. The ideal P in a vector lattice is prime whenever it follows from inf(a,b) E P that at least one of a E P or b E P holds. A principal ideal of a vector lattice E is any ideal genera ted by a singleton {u} denoted by E_u. Ciearly. E_u = {v E E : 3 A > 0 such that |v| 6 A|u|}.

An order closed ideal in a vector lattice is called a band. A band B of a vector lattice E is said to be a projection band if B ф B^d = E where B^d denotes the disjoint complement of B. A vector lattice has the projection property if every band is a. projection band.

Let A Ire a vector lattice and v G A⁺. Then the sequence (a_n)_nGN iii A is calied (v) relatively uniformly convergent to a G A if for every real number e >  0. there exists a ilatural number n_e such that |a_n - a| 6 ev for all n >  n_e. Tins will Ire denoted by an ^ a (v). If an ^ a (v) for some 0 6 v G A. then the sequence (an)_nGN is called ( relatively ) uniformly convergent to a, winch will be denoted by an ^ a (r/uf The notion of (v) ( 'relatively ) uniformly Cauchy sequence is defined in the obvious way. A vector lattice is called (relatively) uniformly complete if every relatively uniformly Cauchy sequence in A has a unique limit. Relatively uniformly limits are unique if A is Archimedean.

Let A and B Ire vector lattices. A multilinear map Ф : An ^ B is said to Ire positive whenever (a1,...,a_n) G (A⁺)n imp lies ^(a1 ,...,a_n) E B+. A multilinear map Ф is said to Ire orthosymmetric if for all (a1,...,a_n) E An such that ai Л aj = 0 for some 1 6 i,j 6 n implies ^(a1,...,a_n) = 0.

• 3.    Main results

We start with some auxiliary results which will be used in the sequel.

Proposition 1. Let A be a vector lattice and n G N. Then the followings are equivalent: 1) A = I1 ф I2 ф ... ф In for some simp)le order ideals I1, I1, . . . , In iii A.

• 2)    A = Rxi ф Rx2 ф ... ф Rxn for some elements x1, x₂,..., x_n G A.

C (1) ^ (2) Siiice Ii is simple for each i = 1, 2 ... ,n. and according to [3. Proposition 1]. it follows that Ii = Rxi for some elements x1 ,x₂,...,x_n G A. Consequently. A = Rx1 ф Rx2 ф ... ф Rxn for sorrie n G N and some elements x1,x₂,... ,x_n G A.

• (2 ) ^ (1) Tins directitni is trivial. B

Definition 1. The depth of a. vector lattice is the supremum (possibly infinite) of the lengths of maximal orthogonal system.

Proposition 2. Let A be a vector lattice. Then the followings are equivalent:

• 1)    A has the projection property and its depth is finite;

• 2)    A = Rxi ф Rx₂ ф ... ф Rxn for some n E N and some elements x₁, x₂,..., x_n E A.

C (1) ^ (2): Let n E N be the depth of A and {e₁, e₂,..., en} be the maximal orthogonal system of length n In A. Since A has the projection property. A = {e₁}^dd ф^}¹¹ ф... {ei}^ddф ... ф {e_n}^dd Let 1 6 i 6 n be fixed and y E {ei}^dd. T1 ien y = y⁺ - y^- wliere y+,y^- E {ei}^dd. Let f₁ = n^y+ydd (ei) and f₂ = n^y+^a (ei) where п^+^аа and n^y+^d are the band projections on {e₁, e₂,..., en} with ranges {y⁺}^dd and {y⁺}^d, respectively. It is easy to see that the system {e₁, e₂,..., e_i-1, f₁, f₂, ei₊₁,..., en} is an orthogonal system of length n +1. Therefore, either fi = 0 о _r ^f2 = 0

Case 1: If f₁ = 0. th en ei = f₂ E {y⁺} - Hence {y⁺} C {ei }^dd C {y⁺}  am I so y⁺ = 0.

Case 2: If /2 = 0. th en ei = f1 E {y+}dd He nee {y-}dd C {ei }dd C {y+}dd C {y-}d and so y- = 0.

Then, the band {ei }^dd Is totally or

{ei}dd = Rei for all 1 6 i 6 n. Then A = Re1 ф Re2 ф ... ф Ren for some n E N.

The Implication (2) ^ (1) is tri vial. B

To clarify next result, we give the following well-known lemma.

Lemma 1. Let A be a universally complet e vector lat tice with a weak order unit e. Then there exists a unique multiplication on A such that A is an f-algebra with e as a unit element.

Definition 2. A vector lattice A is said to be hyper-Archimedean if all quotient spaces E/I, where I is an order ideal in A. are Archimedean.

Several characterizations of hyper-Archimedean vector lattices are known (see, for example, [10], [11, Theorem 37.6, 61.1 and 61.2]). We collect some of them in the following lemma.

Lemma 2. A vector lattice A is hyper-Archimedean if and only if any of the following equivalent conditions holds.

• (i)    Every prime ideal in A is a maximal ideal.

• (ii)    Every ideal in A is uniformly closed.

(hi) The span of the set of all components of u is the principal ideal generated by u for all u E A⁺.

Definition 3. A lattice-ordered algebra A is called ArEman (respectively, super ArEman) if A satisfies the descending chain condition on order ideals (respectively, on bands).

Definition 4. A lattice-ordered algebra A is called Noetherian (respectively, super Noetherian) if A satisfies the ascending chain condition on order ideals (respectively, on bands).

Definition 5. Let A Ire a vector lattice. All element x of A is said to Ire super atomic if the oder ideal Ax genera.l ed by x is of finite dimensional.

We now give the following result which shows the relation between the dimensions of a. vector lattice and its universal completion.

Proposition 3. Let A be a vector lattice and Au be its universal completion. Then A is finite dimension if and only if Au is finite dimension.

C Let A be a finite dimension vector lattice. Since A is finite dimension and the elements of any finite orthogonal system of A are linearly independent, it follows that A has {e1, e2,..., en} as a maximal orthogonal system of length n. Let y E {ei}dd for a fixed index i with 1 6 i 6 n. Then y = y+ - y- with y+,y- E {ei}dd. If y+ = 0 and y- = 0, it is easy to see that the system {e1, e2,..., ei-1, y+, y-, ei+1,..., en} is an orthogonal system of length n + 1. which is a contradiction. Hence. y+ = 0 оr y- = 0. Then, the band {ei}dd is totally ordered for all 1 6 i 6 n. By [3. Pro position 1]. {ei}dd = Rei for all 1 6 i 6 n. Hence, the band generated by ei iii Au will l>e < ?qual to Rei. T1 ien Au = Re1 ф Re2 ф ... ф Ren for some n G N. He nee Au is finite dimension. Conversely, if Au is finite dimension, then so is clearly A. B

Remark 1. If a vector lattice A is finite dimension then A = Au.

We now have all ingredients to give the first main result of this section.

Theorem 3. Let A be a vector lattice and Au be its universal completion. Then the followings are equivalent:

• (1)    A has the projection property and its depth is finite.

• (2)    A = I1 ф I2 ф ... ф In for some simple order ideals I1,I2,... ,I_n in A. and n G N.

• (3)    Au is super Dedekind complete and Artinian.

• (4)    Au is super Dedekind complete and super Artinian.

• (5)    Au is super Dedekind complete and Noetherian.

• (6)    Au is super Dedekind complete and super Noetherian.

• (7)    Au has at least one super atomic weak order unit.

• (8)    A is finite dimension.

C (1) ^ (2) This follows from Propositions 1 and 2.

• (2)    ^ (3) Siiice A = Rx1 ф Rx2 ф ... ф Rxn for some n G N and some elements x1,x₂,... ,x_n G A. it follows that Au = A. Hence, the set of all order ideals of A is finite and then A is Artinian.

(3)^(4) Tins path is trivial.

• (4)    ^ (6) Let (B_n)_nGN be an increasing sequence of bands in Au. Then (ВП)n_GN is a decreasing sequence of bands in Au. By the fact, that A is super Artinian. it follows that there exists n₀ G N such that B^ = B^ for all n > n₀. Consecinently-. Bn = B_n0 for all n > n₀. Therefore Au is super Notherian.

• (6)    ^ (7) Let S = {ei : i G I } be a maximal ortllogonal system in Au. T1 ien e = sup {ei : i G I } is a weak о rder unit of Au. Su ice Au is super Dedekind complete, it follows that there exists an at most countable subset T = {en : n G N} оf S such that e = sup {en : n G N}. Let (B_n)_neN be an increasing se quence of bands in Au where Bn = {V_16i6nei}^dd- Su ice Au is super Notherian, it follows that there exists n₀ G N such that Bn = B_n0 for all n > n₀. Consequently, {V16i6_n0ei}dd = {Vi^^nejdd f°^{r a}H n >  no- Therefore, en = 0 for all n > n₀ and then S is a finite set. Let S = {fjn ,j_n G Jn} be an orthogc>nal system of Au such that en = ^{sup {f}jn ,^jn ^G Jn^} for all ^{1 6 n 6 n}0. Since the set S = ^{j j ^G Jn, ¹ 6 n 6 n0^} is a maximal orthogonal system of Au, it follows that S0 is a finite set. Using the same argument with each fjn, we deduce th at there exist m₀ G N and a maximal orthogonal system K = {kn, 1 6 n 6 m₀} оf Au with e = sup{kn, 1 6 n 6 m₀} such that K will be the finest orthogonal system meaning that we cannot use the decomposition process another time. Consequently, Au = Au ф Au ф ... ф Aum0. Let 1 6 n 6 n₀ and let 0 6 x 6 kn. Let H_kn = {y G Aun ,y = E_16i6m aihi where ai G R and hi is a component of kn}. It is not hard to prove that He is a liyper-Archimodetrn vector sublattice of Au. Su ice K is a finest orthogonal system, it follows that the set of all components of kn is {0, kn}.

By Freudenthal Spectral Theorem, there exists x_m = P_16i6m aihi = amkn with x_n ^ x (r. u). Therefore, there exists an G R such that x = ankn. He nee, H_kn = Aun for all 1 6 n 6 no-

Since any relatively uniformly complete hyper-Archimedean vector lattice is of finite dimension (see [11, Theorems 37.6, 61.4], [12, Theorem 3]), and AUn is relatively uniformly complete, it follows that AU is of finite dimeiision. Therefore, e is super atomic.

• (7)    ^ (8) Lel, e be a super atomic weak order unit of Au. Tlieii AU is of finite dimension. Since x Л ne ^ x (r.u) for all 0 6 x E Au. it foliows that Au is of finite dimension.

• (8)    О (1) Tins equivalence is trivial.

• (1)    ^ (5) Su ice Au is of finite dimension, the set of all order ideals of Au is finite and then Au satisfies the ascending chain eondilion on ideals arid so we are done. B

Let A be a vector lattice and E be a vector sublattice of A. A positive linear operator n : E ^ A is said I,о be a, positive ortho-morphism if n (x) Л y = 0 wlieilever x Л y = 0 for each x,y E E. Ari ortho-morphism, is the difference of two positive orthomorphisms. Orth(E,A) will denote the vector lattice of all orthomorphisms from E to A. Z (E, A) will denote the sublattice of Orth(E, A) consistingg of those n for which there is a non-negalive real A with

—Ax 6 n (x) 6 Ax for all x E E +. Let A be a lattice-ordered algebra. M(A) denotes the set of all maximal twosided algebra ideals. We consider a subset m(A) оf M(A) consisting of relatively uniformly closed ideals.

All prerequisites are made for the second main result of this section.

Theorem 4. Let A be a vector lattice and let Au be its universal completion unital f-algebra. Then the following conditions are equivalent:

• (1)    A is Uni to dimension.

• (2)    Every maximal modular algebra ideal in Au is relatively uniformly closed.

• (3)    Orth(A, Au) = Z (A, Au).

C (1) ^ (2) Sfiice Ais of finite dimension, then Au is of finite dim elision so that Au becomes a, Banach lattice. It is well-known that any maximal modular algebra, ideal of a, commutative Banach algebra, is closed. So we are done.

• (2)    ^ (3) Since any maximal rmrdular algebra, ideal of Au is relatively uniformly closed and by using the main result of [13], we deduce that Orth(Au) = Z(Au). Moreover, it is not fiard l,o prove that Orth(Au) = Orth(A,A^U) aiid Z(A,A^U) = Z(Au).

• (³⁾    ^ (1) since (W) = orthiA, Au). Z(A, A^U) = Z(A^U) „„,1 a-tnAe = Z(A, Au). it follows that Orth(A^U) = Z(Au) = Au. Consequently, Au becomes a Banach lattice and it is well-known that any Banach universally complete vector lattice is of finite dimension. Hence, A is of finite dimonsion and we are done. B

It is well-known that any universally complete vector lattice A is of the form C“ (X) for some Hausdorff extremally disconnected compact topological space X (i. e. the closure of every open set of X is also open). The symbol C “ (X) denotes the collection of all continuous functions f : X ^ x, +w] for which the open sei, dom f = {x E X : —to < f (x) <  +to} is dense in X. Il, is well-known that C“ (X) can be eqmpped with a, unital f-algebra multiplication. Moreover. C“(X) is a Dedekirid complete f-algebra with e := xX as a unit element. The orthomorphisms in C“(X) are the poiiilwiso multiplications, so ^Orth(C“⁽X)) = C“⁽X).           _ _

In order to reach our aim, we need the following;

Theorem 5 [14, 15]. Given an extremally disconnected compact space X, the following properties of a point x E X are pairwise equivalent:

• (1)    The intersection of every sequence of neighborhoods of x is a neighborhood of x.

• (2)    x E dom f for all f E C“(X).

• (3)    If f E C“(X) aiid f (x) = 0. ihen f = 0 in some neighborhood of x.

DEFIXITIOX G. The point, x is called a-isoiated (or a P-poirit. or bounded) whenever x enjoys any of the properties in Theorem 5.

Theorem 6 [14, 15]. The maximal algebra ideals of C“(X ) for x E X are of the form (C“(X))x := {f E C“(X ) : f = 0 in some neig-hborhood of x}.

We now have gathered all ingredient for the third result of this section.

Theorem 7. Let A be a vector lattice and let Au (= C“ (X )) be universal completion unital f-algebra of A. Then the following conditions are equivalent:

• (1)    A is of Unite dimension.

• (2)    Each element of X is o-isolated.

C (1) ^ (2) Siiice A is of finite dimension, then Au is of finite dimension: therefore Au becomes a Вanach unital f-algebra and it is well-known that any maximal algebra ideal of a commutative Banach algebra is closed. By Theorem 6, any maximal algebra ideal is of the form (C“(X))_x for some x E X and since (C^(X))_x (C^(X))_x is relatively uniformly closed, then x is o-isolated (see [14, 15]).

• (1)    ^ (2) Since any point of X is ст-isolated, it follows that any maximal algebra ideal of Au is relatively uniformly closed (see [14, 15]). By the main result of [13], we deduce that Orth(A^u) = Z⁽A^u). Since ^Orth(A^u) = Z⁽A^u) = Au, Au becomes a Banach lattice and it is well-known that any Banach universally complete vector lattice is of finite dimension. Hence. A is of finite dimonsion and we are done. B

Next, we will give a new class of non-finite dimension zero product algebras for which the characterization of Theorem 2 holds.

Theorem 8. Let A be an Archimedean unital f-algebra. Then A is zero product determined if and only if A is hyper-Archimedean.

C For the proof of “only if” part, we will use the same argument as in [16, Theorem 8]. Assume that every bilinear map f : A x A ^ B, where B is an arbitrary vector space over K, with Hie property that for all x,y E A. f (x,y) = 0 wlicricvcr xy = 0. is of tlie form f (x,y) = Ф (xy) for some linear map Ф : A ^ B. Suppose, per contra, that A is riot hyper-Archimedean. It follows that there exists a prime ideal I which is not maximal. Hence the quotient A/I is linearly ordered space that is not isomorphic to R (see [11. Theorem 27.3 ami 33.2]). Let x. y in A/I that are linearly independent. Hence x + y, x are linearly independent. By using Zorn’s Lemma, it is not hard to prove that x, y are contained in a Hamel basis H1 and x + y, x are contained in a Hamel basis H2 such that H1 = H2. Then there exist two linear maps f : A/I ^ R such that f (x) = 1 aiid f (y) = -1 arid g : A/I ^ R such that g(x + y) = 1 and g(y) = -1. Let the bilinear map Ф : A x A ^ R be defined by Ф(a,b) = f (a)g(b). for all a,b E A. Let a, b E A such that ab = 0. Snicc I is a prime ideal, it follows that a E I or b E I. Hence a = 0 оr b = 0. Consecinently. f (a) = 0 оr g(b) = 0. Therefore Ф(a, b) = 0. Непсе. Ф is ortliosymmetric. Whereas. Ф(x,y) = f (x)g(y ) = f (y)g(x) = Ф(y,x). Tliat is Ф is riot symmetric. Непсе Ф is riot, of the form Ф(xy) for some linear map Ф : A ^ R. which is a. contradiction.

Then “if” part remains. We will use the same argument as in [17, Theorem 1]. Let Ф: A x A ^ B. where B is an arbitrary r-ector space over K. with the property that for all x, y E A. Ф(x, y) = 0 wherlever xy = 0.

Let x,y E A. It follows that x = Pⁿ=1 ai ei and y = P^m=1 ej fj, where ei and fj are components of e = |x| + |y|. Then

^Ф(x,У⁾= X ^ai^ejWi ^,fj ) •                         (1)

16i6n, 16j 6m

Let ed be the disjoint complement of ei. He nee e = ei + ed where ei Л ed = 0. Then fj = fj Л e = fj Л (ei + ed) = (fj Л ei) + (fj Л ed).

Since (fj Л ed) Л ei = 0. then (fj Л ed) ei = 0. Consequently.

^Ф(ei^,fj) = ^Ф(^ei, (^fj ^Л ei) ⁺ (^fj ^{Л e}i )) = MeiJj ^Л ei )•

Moreover ei = ei Л e = ei Л (fj + j = (fj Л ei ) + (ei Л fjd). Hence.

WiJj ) = ^(ei, fj Л ei) = Ф((fj Л ei) + (ei Л fd), fj Л ei ).

Since (ei Л fjd) Л (fj Л ei) = 0, it folloxvs that (ei Л fjd) (fj Л ei) = 0 and then ^ф(ei ^,fj ) = ^Ф(Л ^Л ei^,fj ^Л ei ) = ^^(fj ^,ei)

By using the same argument, we prove that

Ф(в» ,fj ) = Ф(ei,fj Л ei ) = ^(fj Л ei,fj Л ei) = ^(fj,ei).

Therefore, in view of equality (1), we have

Ф(x,y) = Ф(y,x).

Let u be the unit of A and let z E A. Then the following bilinear map Ф_z : A x A ^ B defined by Ф_z (x,y) = Ф(xz,y). for all x,y E A, satisfies the iproperty that Ф(x,y) = 0 whenever xy = 0. Therefore, by using the argument as for Ф. we deduce that Ф_z (x, y) = Ф_z (y,x) = ф_х (z,y) = ф_х (y,z). for all x,y E A. In particular if z = u, it follows that ф (x, y) = ф_х (y, e) = Ф (xy, e). Consequently. Ф is not of the form Ф (xy) . xvhmc Ф : A ^ B is defined by Ф (x) = Ф (x, e). for all x E A and we a re done. B

It should be noted that a relatively uniformly complete unital hyper-Archimedean f-al-gebra is of finite dimension (see [11, Theorems 37.6, 61.4], [12, Theorem 3]). Consequently, we have the following characterization.

Theorem 9. Let A be a relatively uniformly unital f-algebra. Then the following properties are equivalent.

• (1)    A is zero product determined.

• (2)    A is hyper-Archimcdcan.

• (3)    A is of finite dimension.