Research Article | Open Access

F. B. H. Jamjoom, "On the Tensor Products of Maximal Abelian -Algebras", *International Journal of Mathematics and Mathematical Sciences*, vol. 2009, Article ID 621386, 7 pages, 2009. https://doi.org/10.1155/2009/621386

# On the Tensor Products of Maximal Abelian -Algebras

**Academic Editor:**Jie Xiao

#### 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.

#### References

- H. Hanche-Olsen, βOn the structure and tensor products of $JC$-algebras,β
*Canadian Journal of Mathematics*, vol. 35, no. 6, pp. 1059β1074, 1983. View at: Google Scholar | MathSciNet - H. Hanche-Olsen and E. Størmer,
*Jordan Operator Algebras*, Pitman, 1984. - L. J. Bunce and J. D. M. Wright, βIntroduction to the $K$-theory of Jordan ${C}^{\ast}$-algebras,β
*The Quarterly Journal of Mathematics. Oxford Second Series*, vol. 40, no. 160, pp. 377β398, 1989. View at: Publisher Site | Google Scholar | MathSciNet - E. Størmer, βJordan algebras of type $I$,β
*Acta Mathematica*, vol. 115, pp. 165β184, 1966. View at: Publisher Site | Google Scholar | MathSciNet - J. D. M. Wright, βJordan ${C}^{\ast}$-algebras,β
*The Michigan Mathematical Journal*, vol. 24, no. 3, pp. 291β302, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. V. Kadison and J. R. Ringrose,
*Fundamentals of the Theory of Operator Algebras I*, vol. 100 of*Pure and Applied Mathematics*, Academic Press, New York, NY, USA, 1983. View at: MathSciNet - M. Takesaki,
*Theory of Operator Algebras. I*, Springer, New York, NY, USA, 1979. View at: MathSciNet - R. V. Kadison and J. R. Ringrose,
*Fundamentals of the Theory of Operator Algebras II*, vol. 100 of*Pure and Applied Mathematics*, Academic Press, New York, NY, USA, 1986. View at: MathSciNet - D. M. Topping, βJordan algebras of self-adjoint operators,β
*Memoirs of the American Mathematical Society*, vol. 53, p. 48, 1965. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. B. H. Jamjoom, βOn the tensor products of $JW$-algebras,β
*Canadian Journal of Mathematics*, vol. 47, no. 4, pp. 786β800, 1995. View at: Google Scholar | MathSciNet - P. M. Cohn,
*Algebra II*, John Wiley & Sons, Chichester, UK, 2nd edition, 1989. View at: MathSciNet - F. B. H. Jamjoom, βThe connection between the universal enveloping ${C}^{\ast}$-algebra and the universal enveloping von Neumann algebra of a JW-algebra,β
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 112, no. 3, pp. 575β579, 1992. View at: Publisher Site | Google Scholar | MathSciNet - F. B. H. Jamjoom, βOn the tensor products of simple $JC$-algebras,β
*The Michigan Mathematical Journal*, vol. 41, no. 2, pp. 289β295, 1994. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2009 F. B. H. Jamjoom. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.