Maximal discrete subgroups in unitary groups of operator algebras

TL;DR

This paper demonstrates the existence of maximal discrete subgroups in the projective unitary groups of group von Neumann algebras for certain groups, using elementary methods and free probability theory.

Contribution

It establishes the presence of maximal discrete subgroups in these unitary groups for mixed-identity-free groups, expanding understanding of their structure.

Findings

Maximal discrete subgroups exist in the projective unitary groups of group von Neumann algebras.

Elementary proofs are provided using free probability theory.

The paper clarifies the situation for C*-algebras.

Abstract

We show that if a group G is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for C*-algebras.