The splitting problem for subspaces of tensor products of operator algebras

The main result of this paper is that if is a von Neumann algebra that is a factor and has the weak* operator approximation property (the weak* OAP), and if is a von Neumann algebra, then every -weakly closed subspace of that is an -bimodule (under multiplication) splits, in the sense that there is a -weakly closed subspace of such that . Note that if is a von Neumann subalgebra of , then is an -bimodule if and only if . So this result is a generalization (in the case where has the weak* OAP) of the result of Ge and Kadison that if is a factor, then every von Neumann subalgebra of that contains splits. We also obtain other results concerning the splitting of -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.