On unitary groups of crossed product von Neumann algebras

TL;DR

This paper investigates conditions under which unitary subgroups of crossed product von Neumann algebras can be conjugated into a specific subgroup, generalizing previous results from group von Neumann algebras to more general crossed products.

Contribution

It provides a new sufficient condition for conjugating unitary subgroups into a canonical subgroup in crossed product von Neumann algebras, extending prior work.

Findings

Established a general criterion for conjugation of unitary subgroups.

Extended results from group von Neumann algebras to crossed products.

Generalized prior theorems by Ioana, Popa, and Vaes.

Abstract

We consider the tracial crossed product algebra $M=A\rtimes\Lambda$ arising from a trace preserving action $\sigma:\Lambda \curvearrowright A$ of a discrete group $\Lambda$ on a tracial von Neumann algebra $A$. For a unitary subgroup $\mathcal{G}\subset \mathcal{U}(M)$, we study when this $\mathcal{G}$ can be conjugated into $\mathcal{U}(A)\cdot\Lambda$ in $M$. We provide a general sufficient condition for this to happen. Our result generalizes a result of Ioana, Popa and Vaes, which treats the case when $M$ is the group von Neumann algebra $L(\Lambda)$.