On maximal injective subalgebras of factors (Q1915383)

The main subject of this paper is the existence of a splitting property of subalgebras of tensor products of von Neumann algebras. Related to this question is the description of maximal injective subalgebras of a von Neumann algebra and, in particular, of maximal injective subfactors of a factor. The splitting problem referred to is the following: when is a subalgebra \(S\) of the tensor product of \(A_1\) and \(A_2\) the tensor product of subalgebras \(S_1\) of \(A_1\) and \(S_2\) of \(A_2\). The main result says that a von Neumann subalgebra \(S\) of the tensor product of a factor of finite type \(M\) and a finite von Neumann algebra \(R\) splits as the tensor product of \(M\) and a von Neumann subalgebra of \(R\) provided that \(S\) contains \(M\otimes \mathbb{C} I\). The reader may consult related results in the paper ``On tensor products of von Neumann algebras'', Invent. Math. 123, No. 3, 453-466 (1996), by the author and \textit{R. V. Kadison}.