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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The splitting problem for subspaces of tensor products of operator algebras
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by Jon Kraus PDF
Proc. Amer. Math. Soc. 132 (2004), 1125-1131 Request permission

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
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1125-1131
  • MSC (2000): Primary 46L10
  • DOI: https://doi.org/10.1090/S0002-9939-03-07243-5
  • MathSciNet review: 2045429