A random matrix approach to absorption in free products

TL;DR

This paper introduces a free entropy approach to absorption phenomena in free product von Neumann algebras, proving maximal amenability of generator MASA and unifying results via 1-bounded entropy.

Contribution

It provides the first free entropy proof of Popa's maximal amenability result and offers a unified framework for amenable absorption using 1-bounded entropy and random matrix models.

Findings

Proves generator MASA in free group factors is maximal amenable.

Shows subalgebras with zero 1-bounded entropy are absorbed in free products.

Unifies various absorption results through random matrix models with exponential concentration.

Abstract

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $\mathcal{P} \leq \mathcal{M}$ to have this absorption property, it suffices for $\mathcal{M}$ to admit random matrix models that have exponential concentration of measure and that "simulate" the conditional expectation onto $\mathcal{P}$.