Bass-Serre rigidity results in von Neumann algebras

TL;DR

This paper establishes new rigidity results for von Neumann algebras derived from free ergodic actions of free product groups, demonstrating the primeness of certain amalgamated free product factors in both type II_1 and type III cases.

Contribution

It introduces novel Bass-Serre type rigidity results for von Neumann algebras associated with free product group actions and proves primeness of specific amalgamated free product factors.

Findings

Non-amenable factors from amalgamated free products are prime.

New examples of prime factors in type II_1 and type III.

Rigidity results connect group actions with von Neumann algebra structure.

Abstract

We obtain new Bass-Serre type rigidity results for ${\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras $\mathcal{M}_1 \ast_B \mathcal{M}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\rm II_1}$ and in the type ${\rm III}$ case, new examples of prime factors.