International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 621386 | 7 pages | https://doi.org/10.1155/2009/621386

On the Tensor Products of Maximal Abelian J W -Algebras

Academic Editor: Jie Xiao
Received24 Apr 2009
Accepted07 Jun 2009
Published10 Aug 2009

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    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  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 [1–5] 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 [6–8].

A projection of a -algebra is said to be  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   if there is a family of abelian projections such that the central support of in equals the unit of M, and card (see [1, Section  5.3]).   A 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  6.1]).

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  2.1]. A -algebra is universally reversible if and only if it has no spin factor representations other than onto and [2, Theorems  2.2]. 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  11.2.18] for von Neumann tensor products (cf. Example 2.2). Then we prove that a Jordan analog of Theorem  11.2.18 in [6] can be established in some particular cases.

Theorem 1.1 (see [9, Proposition  1]). 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. Let    and    be a pair of -algebras canonically embedded in their respective universal enveloping von Neumann algebras   and . Then the -tensor product    of and   is 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,Theorem  2.9]). Let    and  be  - algebras.  If    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.   Indeed, let be a -subalgebra of a spin factor . If then is the only projection in ,    since every projection in is minimal, and hence  .   If , then any family of orthogonal central projections of   contains at most two projections. Indeed if , then .   Since is a minimal projection, we see that one of    must be zero. It is clear that if  is a factor, then it is of Type  , and hence it is a spin factor. Recall that , where is the 4-dimensional spin factor   [1, 6.2.1]: which is an 8-dimensional real  -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  7.5]). On the other hand, is universally reversible, by [10, Proposition  2.7] 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   is a maximal abelian subalgebra of , then   is a maximal associative-subalgebra of ,  since .

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  generated by  .

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  3.2], 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   be a maximal associative -subalgebra of a -algebra Suppose that is isomorphic to the self adjoint part   of a von Neumann algebra and has no one-dimensional representations. Then is not a maximal abelian on Neumann subalgebra of  .

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

Lemma 2.6. Let   be a maximal associative -subalgebra of a -algebra . If is -isomorphic to a complex -algebra,   then    is not a maximal abelian von Neumann subalgebra of  .

Proof. Since is the complex -algebra generated by in [12, Theorem  2.7], is the wea-closure of in . Therefore, is a complex -algebra, which implies that for some norm closed ideal of isomorphic to [13, Lemma  1], 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  1.5.4], 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. Let    be a universally reversible -algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of type  . If    is a maximal associative subalgebra of  ,  then is a maximal abelian von Neumann subalgebra of   .

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  1]. Hence , since   is 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  2.6].

Proposition 2.9. Let   be a maximal associative subalgebra of a -algebra  , and suppose that    is universally reversible   -algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of type  . Then is a maximal associative -subalgebra of  .

Proof. Note first that is a von Neumann -subalgebra of [8, Theorem  11.2.10], and is a -subalgebra of , since . By Proposition  5.2, is maximal abelian in , and hence, is maximal abelian in [8, Corollary  11.2.18] and [10, Theorem  2.9]. The result is now obvious, since and [10, Theorem  2.9].

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

Proof. Let be the decomposition of into abelian part and nonabelian part . Then , where is obviously a maximal associative subalgebra of . By [10, Remark  2.14], It is clear now that is a maximal associative -subalgebra of , since is obviously associative, and is maximal in  , 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 of   and , respectively, then is a maximal associative -subalgebra of  .

Proof. Let be the decomposition of into abelian parts , and nonabelian parts . Then and , where and . Therefore, by [13, Remark  2.14]. The proof is complete, by Propositions 2.9 and 2.10.

References

  1. H. Hanche-Olsen, β€œOn the structure and tensor products of JC-algebras,” Canadian Journal of Mathematics, vol. 35, no. 6, pp. 1059–1074, 1983. View at: Google Scholar | MathSciNet
  2. H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Pitman, 1984.
  3. L. J. Bunce and J. D. M. Wright, β€œIntroduction to the K-theory of Jordan C-algebras,” The Quarterly Journal of Mathematics. Oxford Second Series, vol. 40, no. 160, pp. 377–398, 1989. View at: Publisher Site | Google Scholar | MathSciNet
  4. E. Størmer, β€œJordan algebras of type I,” Acta Mathematica, vol. 115, pp. 165–184, 1966. View at: Publisher Site | Google Scholar | MathSciNet
  5. J. D. M. Wright, β€œJordan C-algebras,” The Michigan Mathematical Journal, vol. 24, no. 3, pp. 291–302, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. 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
  7. M. Takesaki, Theory of Operator Algebras. I, Springer, New York, NY, USA, 1979. View at: MathSciNet
  8. 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
  9. D. M. Topping, β€œJordan algebras of self-adjoint operators,” Memoirs of the American Mathematical Society, vol. 53, p. 48, 1965. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. F. B. H. Jamjoom, β€œOn the tensor products of JW-algebras,” Canadian Journal of Mathematics, vol. 47, no. 4, pp. 786–800, 1995. View at: Google Scholar | MathSciNet
  11. P. M. Cohn, Algebra II, John Wiley & Sons, Chichester, UK, 2nd edition, 1989. View at: MathSciNet
  12. F. B. H. Jamjoom, β€œThe connection between the universal enveloping C-algebra and the universal enveloping von Neumann algebra of a JW-algebra,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 112, no. 3, pp. 575–579, 1992. View at: Publisher Site | Google Scholar | MathSciNet
  13. F. B. H. Jamjoom, β€œOn the tensor products of simple JC-algebras,” The Michigan Mathematical Journal, vol. 41, no. 2, pp. 289–295, 1994. View at: Publisher Site | Google Scholar | MathSciNet

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.


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