On a Question of D. Shlyakhtenko

TL;DR

This paper constructs two specific infinite conjugacy class groups with free, ergodic, measure-preserving actions that are orbit equivalent, yet their group von Neumann algebras are not stably isomorphic, highlighting nuanced differences in group actions and algebraic structures.

Contribution

It provides explicit examples of groups with orbit equivalent actions but non-isomorphic von Neumann algebras, addressing a question posed by D. Shlyakhtenko.

Findings

Existence of two groups with orbit equivalent actions

Their von Neumann algebras are not stably isomorphic

Highlights subtle distinctions in group action and algebraic properties

Abstract

In this short note we construct two countable, infinite conjugacy class groups which admit free, ergodic, probability measure preserving orbit equivalent actions, but whose group von Neumann algebras are not (stably) isomorphic.