The splitting problem for subspaces of tensor products of operator algebras

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.