Bass-Serre rigidity results in von Neumann algebras

Abstract

We obtain new Bass-Serre-type rigidity results for ${\mathrm{II}}_1$ 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 nonamenable factor arising as an amalgamated free product of von Neumann algebras $M_1{*}_B{M}_2$ over an abelian von Neumann algebra $B$ is prime, that is, cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\mathrm{II}}_1$ and in the type $\mathrm{III}$ cases, new examples of prime factors.