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The splitting problem for subspaces of tensor products of operator algebras


Author: Jon Kraus
Journal: Proc. Amer. Math. Soc. 132 (2004), 1125-1131
MSC (2000): Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-03-07243-5
Published electronically: November 4, 2003
MathSciNet review: 2045429
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Abstract: The main result of this paper is that if $N$ is a von Neumann algebra that is a factor and has the weak* operator approximation property (the weak* OAP), and if $R$ is a von Neumann algebra, then every $\sigma $-weakly closed subspace of ${{N}\bar \otimes {R}}$that is an ${N}\bar \otimes {\mathbb{C} 1_{R}}$-bimodule (under multiplication) splits, in the sense that there is a $\sigma $-weakly closed subspace $T$ of $R$ such that $S={{N}\bar \otimes {T}}$. Note that if $S$ is a von Neumann subalgebra of ${{N}\bar \otimes {R}}$, then $S$is an ${N}\bar \otimes {\mathbb{C} 1_{R}} $-bimodule if and only if ${N}\bar \otimes {\mathbb{C} 1_{R}} \subset S$. So this result is a generalization (in the case where $N$ has the weak* OAP) of the result of Ge and Kadison that if $N$ is a factor, then every von Neumann subalgebra $M$ of ${{N}\bar \otimes {R}} $ that contains ${N}\bar \otimes {\mathbb{C} 1_{R}}$splits. We also obtain other results concerning the splitting of $\sigma $-weakly closed subspaces of tensor products of von Neumann algebras and the splitting of normed closed subspaces of C*-algebras that generalize results previously obtained for von Neumann subalgebras and C*-subalgebras.


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Additional Information

Jon Kraus
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260-2900
Email: mthjek@acsu.buffalo.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07243-5
Received by editor(s): June 14, 2002
Received by editor(s) in revised form: December 13, 2002
Published electronically: November 4, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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