Gamma stability in free product von Neumann algebras

Abstract

Let $(M, \varphi) = (M_1, \varphi_1) \ast (M_2, \varphi_2)$ be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer $M_1^{\varphi_1}$ is diffuse. We first show that any intermediate subalgebra $M_1 \subset Q \subset M$ which has nontrivial central sequences in $M$ is necessarily equal to $M_1$. Then we obtain a general structural result for all the intermediate subalgebras $M_1 \subset Q \subset M$ with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III$_1$ factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.