#### Abstract

It is well known in the work of Kadison and Ringrose (1983)that if and are maximal abelian von Neumann subalgebras of von Neumann algebras and , respectively, then is a maximal abelian von Neumann subalgebra of . It is then natural to ask whether a similar result holds in the context of -algebras and the -tensor product. Guided to some extent by the close relationship between a -algebra M and its universal enveloping von Neumann algebra , we seek in this article to investigate the answer to this question.

#### 1. Introduction

A -*algebra*βββ is a norm (uniformly) closed Jordan subalgebra of the Jordan algebra of all bounded self adjoint operators on a Hilbert space The Jordan product is given by . A subspace of a -algebra is called *a Jordan ideal* if for every and every . A -algebra is said to be *simple* if it has no nontrivial norm closed Jordan ideals. A -*algebra*β is a weakly closed -algebra. If is a -algebra (resp., -algebra), let (resp., ) be the universal enveloping -algebra ** (**resp., von Neumann algebra) of , and let (resp., ) be the canonical involutive *-antiautomorphism of (resp., ). Usually we will regard as a generating Jordan subalgebra of ) and so that and fix each point of . The real -algebra satisfies
and the real von Neumann algebra satisfies
The reader is refered to [1β5] for a detailed account of the theory of -algebras and -algebras. The relevant background on the theory of -algebras and von Neumann algebras can be found in [6β8].

A projection of a -algebra is said to beββ*abelian* if is associative, and it is called *minimal* if it is nonzero and contains no other nonzero projections of , or equivalently, is minimal if and only if . *A **-factor* is a -algebra with trivial centre; a Type **-***factor* is a -factor which contains a minimal projection. A -*algebra* is said to be of *Type*ββ if there is a family of abelian projections such that the central support of in equals the unit of M, and card (see [1, Sectionββ5.3]). β*βA spin factor * is a real Jordan algebra with identity , where is a real Hilbert space of dimension at least two. The Jordan product on is defined by
and the norm on is given by
A spin factor is universally reversible when or , nonreversible when or , and it can be either reversible or nonreversible when . A spin factor is a simple reflexive -algebra and constitutes the Type -factor (see [2, Section β6.1]).

A linear map between -algebras and is called *a (Jordan) homomorphism* if it preserves the Jordan product. A Jordan homomorphism which is one to one is called *a Jordan isomorphism*. *A factor representation* of a -algebra is a (Jordan) homomorphism of onto a weakly dense subalgebra of a -factor . *Type I factor representations* are defined accordingly.

A -algebra is said to be *reversible* if whenever and is said to be *universally reversible* if is reversible for every representation of [2, page 5]. The only universally reversible spin factors are and [2, Theorem β2.1]. A -algebra is universally reversible if and only if it has no spin factor representations other than onto and [2, Theorems β2.2]. Every -algebra without a direct summand of Type is universally reversible [1, 5.1.5, 5.3.5, 6.2.3].

Two elements and of a -algebra are said to *operator commute* if , where is the multiplication operator defined by , for all . A -algebra is called *associative* if all its elements operators commute. A -subalgebra of a -algebra is called *maximal associative* if it is not contained in any larger associative -subalgebra of . If is a -subalgebra of a -algebra and is the set of all elements of which operator commutes with all elements of , then is a maximal associative -subalgebra of if and only if . Indeed, since is associative, and together with any element of generates an associative -subalgebra of which implies that since is maximal abelian. In particular, if is an associative -algebra, then is maximal associative if and only if .

This article aims to study the relationship between the maximality of an associative -subalgebra of a -algebra and that of in . We give a counterexample which rules out the establishing of a result in the theory of -tensor products analog to that given in [6,Theorem β11.2.18] for von Neumann tensor products (cf. Example 2.2). Then we prove that a Jordan analog of Theorem β11.2.18 in [6] can be established in some particular cases.

Theorem 1.1 (see [9, Proposition β1]). *Let be a -algebra, and let . Then the following are equivalent:*(i)(ii)(iii)*That is, and operators commute if and only if they commute under ordinary operator multiplication.*

*Definition 1.2. *Letββββandββββbe a pair of -algebras canonically embedded in their respective universal enveloping von Neumann algebras ββand *. *Then the -tensor productββββof and ββis the -algebra generated by in . The reader is referred to [10] for the properties of the -tensor product of -algebras.

Theorem 1.3 (see [10,Theoremβ 2.9]). *Letββββandββbeββ- algebras.ββIfβββ is universally reversible, then
*

#### 2. Maximal Abelian -algebras

Let and be maximal abelian von Neumann subalgebras of von Neumann algebras and , respectively, then is a maximal abelian von Neumann algebra of (see [6, 11.2.18]). In Example 2.2, we show that the Jordan analog of this result, in the context of -algebras and the -tensor product, is not true in general. However, it is shown in Theorem 2.11 that the result does hold in special circumstances. *Remark 2.1. * Note that any -subalgebra of a spin factor which is not a spin factor is of dimension at most *2.*ββ Indeed, let be a -subalgebra of a spin factor *.* If then is the only projection inβ*, *ββ since every projection in is minimal, and henceββ*.* ββIf *, * then any family of orthogonal central projections of ββcontains at most two projections. Indeed if *,* then *.* β Since is a minimal projection, we see that one of ββ must be zero. It is clear that ifββis a factor, then it is of Typeββ*,* and hence it is a spin factor. Recall that *,* where is the 4-dimensional spin factor ββ[1, 6.2.1]:
which is an 8-dimensional realββ*-*algebra.

*Example 2.2. *Let be a maximal abelian -subalgebra of Then is not a maximal abelian subalgebra of *. *

*Proof. *By the above remark, , and hence for some minimal projections . Therefore,
and hence , since *Β·* (see [11, Corollary β7.5]). On the other hand, is universally reversible, by [10, Proposition β2.7] which implies that
since [3, page 385]. It can be seen that a maximal abelian -subalgebra of is of dimension 8, which implies that is not maximal abelian in *Remark 2.3. *Note that if is an associative -subalgebra of a -algebra such that ββis a maximal abelian subalgebra of , then ββis a maximal associative-subalgebra ofβ*,*ββsince .

Lemma 2.4. *Let be an associative -subalgebra of a -algebra . Then,
**
is an abelian von Neumann algebra, where is the -closure of the -subalgebra of βgenerated by β.*

*Proof. *Being associative, has no representation into a spin factor of the form and is, therefore, universally reversible. It follows from [3, page 383] that
Therefore, by [3, Corollary β3.2], is isomorphic to the wea-closure of the real -subalgebra of generated by , and the result follows.

Recall that if is a -algebra isomorphic to the self adjoint part of a von Neumann algebra and has no one-dimensional representations, then is -isomorphic to , where is the opposite algebra of [2, 7.4.15]. A real -algebra can be realized as a complex -algebra if there is a -algebra isomorphism of a complex -algebra onto . In this case, the real linear isometry on defined, for each in by is such that and coincide.

Lemma 2.5. *Let ββbe a maximal associative -subalgebra of a -algebra Suppose that is isomorphic to the self adjoint part ββof a von Neumann algebra and has no one-dimensional representations. Then is not a maximal abelian on Neumann subalgebra of β.*

*Proof. *Identifying with , is a von Neumann subalgebra of both and , and hence, the von Neumannβ subalgebra of is abelian and contains , which implies that ββis not maximal abelian in .

Lemma 2.6. *Let ββbe a maximal associative -subalgebra of a -algebra . If is -isomorphic to a complex -algebra, ββthenββββis not a maximal abelian von Neumann subalgebra of β.*

*Proof. *Since is the complex -algebra generated by in [12, Theorem β2.7], is the wea-closure of in . Therefore, is a complex -algebra, which implies that for some norm closed ideal of isomorphic to [13, Lemma β1], so that , where is the wea-closure of in . Hence, is isomorphic to . Let be the isomorphism of onto , and let be the corresponding real linear operator on , defind above. Then, using Lemma 2.4, there exists an isomorphism from the -algebra into such that, for elements and in ,
It follows that and are -isomorphisms of into and , respectively. Since a -isomorphism between -algebras is an isometry [7, Corollary β1.5.4], we may identify with and . It follows that is an abelian von Neumann subalgebra of , proving that is not maximal abelian in .

Proposition 2.7. *Letββββbe a universally reversible -algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of typeββ. Ifββββis a maximal associative subalgebra ofββ,ββthen is a maximal abelian von Neumann subalgebra of ββ.*

*Proof. *By Lemma 2.4, . If is not maximal abelian in , there exists an element , such that together with generate an abelian von Neumann subalgebra of . Let , . Since , then either or (or both) does not belong to . Suppose that , since , then , for some . Then either or (or both) does not belong to . Since , we have , and , and so , since is a universally reversible [3, page 383]. Therefore, must be the zero element, since it obviously commutes with all elements in . On the other hand, . Since for all , for all , and so and operators commute relative to the Jordan product in [9, Proposition β1]. Hence , since ββis a maximal associative subalgebra of , which implies that . Therefore, , a contradiction. This proves the result.

Lemma 2.8. *Let be a universally reversible -algebra not isomorphic to the self adjoint part of a von Neumann algebra. If is a maximal associative subalgebra of , then is a maximal abelian von Neumann subalgebra of .*

*Proof. *Splitting as the direct sum of a -algebra of type (the abelian part) and a -algebra without direct summands of type (the nonabelian part). It is clear that is a maximal associative subalgebra of and . By Proposition 2.7, is a maximal abelian von Neumann subalgebra of , and hence is a maximal abelian von Neumann subalgebra of , since [12, Lemma β2.6].

Proposition 2.9. *Let ββbe a maximal associative subalgebra of a -algebraββ, and suppose thatβ ββis universally reversibleββ -algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of typeββ. Then is a maximal associative -subalgebra ofββ.*

*Proof. *Note first that is a von Neumann -subalgebra of [8, Theorem β11.2.10], and is a -subalgebra of , since . By Proposition β5.2, is maximal abelian in , and hence, is maximal abelian in [8, Corollary β11.2.18] and [10, Theorem β2.9]. The result is now obvious, since and [10, Theorem β2.9].

Proposition 2.10. *Let be an associative -algebra, and let βbe a universally reversible -algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of typeββ. If ββis a maximal associative subalgebra ofββ, ββthen is a maximal associative -subalgebra ofββ.*

*Proof. *Let be the decomposition of into abelian part and nonabelian part . Then , where is obviously a maximal associative subalgebra of . By [10, Remark β2.14],
It is clear now that is a maximal associative -subalgebra of , since is obviously associative, and is maximal inββ, by Proposition 2.9.Theorem 2.11. *Let and be universally reversible -algebras not isomorphic to the self adjoint parts of von Neumann algebras. If and are maximal associative subalgebra ofββ and , respectively, then is a maximal associative -subalgebra ofββ.*

*Proof. *Let be the decomposition of into abelian parts , and nonabelian parts . Then and , where and . Therefore,
by [13, Remark β2.14]. The proof is complete, by Propositions 2.9 and 2.10.