## The splitting problem for subspaces of tensor products of operator algebras

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- by Jon Kraus
- Proc. Amer. Math. Soc.
**132**(2004), 1125-1131 - DOI: https://doi.org/10.1090/S0002-9939-03-07243-5
- Published electronically: November 4, 2003
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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.## References

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## Bibliographic 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