W*-superrigidity for cocycle twisted group von Neumann algebras

TL;DR

This paper establishes new W*-superrigidity results for certain cocycle-twisted group von Neumann algebras, showing that algebraic isomorphisms imply virtual isomorphisms of the underlying groups, even with twists.

Contribution

It introduces groups with strong W*-superrigidity properties allowing for arbitrary 2-cocycle twists, expanding the class of known superrigid von Neumann algebras.

Findings

Groups with W*-superrigidity under cocycle twists constructed

Virtual isomorphism of von Neumann algebras implies group virtual isomorphism

Examples of indecomposable II₁ factors not arising from twisted groupoid algebras

Abstract

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups $G$ and $\Lambda$ must be virtually isomorphic. Moreover, we allow both group von Neumann algebras to be twisted by an arbitrary $2$-cocycle. We also give examples of II$_1$ factors $N$ that are indecomposable in every sense: they are not virtually isomorphic to any cocycle twisted groupoid von Neumann algebra.