Asymptotic structure of free product von Neumann algebras

TL;DR

This paper investigates the asymptotic structure of free product von Neumann algebras, showing that certain subalgebras with diffuse properties are contained within the original component algebras, thus resolving key questions about their maximal amenability.

Contribution

It generalizes previous results and fully addresses the maximal amenability and property Gamma questions for free product von Neumann algebras.

Findings

Subalgebras with diffuse properties sit inside the original algebra component.

The results extend previous work and settle longstanding open questions.

Provides a comprehensive understanding of the asymptotic structure of free product von Neumann algebras.

Abstract

Let $(M, \varphi) = (M_1, \varphi_1) \ast (M_2, \varphi_2)$ be the free product of any $\sigma$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$ and the central sequence algebra $Q' \cap M^\omega$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1 \subset M$ in arbitrary free product von Neumann algebras.