A remark on amenable von Neumann subalgebras in a tracial free product

TL;DR

This paper proves that in a nontrivial tracial free product of finite von Neumann algebras, any amenable subalgebra intersecting diffusely with one component must be contained within that component, refining previous results.

Contribution

It establishes a new rigidity result for amenable subalgebras in free product von Neumann algebras, extending Houdayer's earlier work.

Findings

Amenable subalgebras with diffuse intersection are contained in one component

Refines understanding of subalgebra structure in free product von Neumann algebras

Extends previous results by Houdayer

Abstract

Let $M=M_1*M_2$ be a nontrivial tracial free product of finite von Neumann algebras. We prove that any amenable subalgebra of $M$ that has a diffuse intersection with $M_1$ is in fact contained in $M_1$. This has been proved by C. Houdayer under more general circumstances.