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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On maximal injective subalgebras


Author: Mingchu Gao
Journal: Proc. Amer. Math. Soc. 138 (2010), 2065-2070
MSC (2010): Primary 46L10
Published electronically: January 7, 2010
MathSciNet review: 2596043
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Abstract: Let $ \mathcal{A}_i$ be a type $ I$ von Neumann subalgebra in a type $ II_1$ factor $ \mathcal{M}_i$ with the faithful trace $ \tau_i$ such that $ \mathcal{A}_i'\cap\mathcal{M}_i\subseteq \mathcal{A}_i$, for $ i=1, 2, \cdots$. Moreover, suppose $ \mathcal{A}_i$ has the asymptotically orthogonal property in $ \mathcal{M}_i$ after tensoring the finite von Neumann algebra $ \otimes_{j\ne i}\mathcal{M}_j$, for all $ i=1,2,\cdots$. Then we show that $ \otimes_{i=1}^\infty\mathcal{A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $ \otimes_{i=1}^\infty\M_i$. As a consequence, we get the following result. Let $ \{\mathbb{F}_{n_i};i=1,2, \cdots\}$ be a sequence of free groups with $ n_i$ ($ >1$) generators. Let $ \mathcal{A}_i$ be the masa of group von Neumann algebra $ \mathcal{L}_{\mathbb{F}_{n_i}}$ generated by a generator of $ \mathbb{F}_{n_i}$ or by the sum of all generators and their inverses of the group. Then $ \otimes_{i=1}^\infty\mathcal{A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $ \otimes_{i=1}^\infty\mathcal{L}_{\mathbb{F}_{n_i}}$.


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Additional Information

Mingchu Gao
Affiliation: Department of Mathematics, Louisiana College, Pineville, Louisiana 71359
Email: gao@lacollege.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10219-6
PII: S 0002-9939(10)10219-6
Keywords: Finite von Neumann algebras, maximal injective subalgebras, tensor products, free group von Neumann algebras.
Received by editor(s): March 16, 2009
Received by editor(s) in revised form: September 20, 2009
Published electronically: January 7, 2010
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.