# 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 algebrasHTML articles powered by AMS MathViewer

by Jon Kraus
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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