C*-simplicity of relative profinite completions of generalized Baumslag-Solitar groups

TL;DR

This paper extends the understanding of C*-simplicity to certain locally compact groups, specifically generalized Baumslag-Solitar groups, by establishing sufficient conditions for their C*-simplicity and uniqueness of KMS-weights.

Contribution

It generalizes previous results on Baumslag-Solitar groups to a broader class of groups called generalized Baumslag-Solitar groups, providing new criteria for C*-simplicity.

Findings

Established sufficient conditions for C*-simplicity of these groups.

Proved the uniqueness of KMS-weights for their reduced group C*-algebras.

Extended known constructions of non-discrete C*-simple groups.

Abstract

Suzuki recently gave constructions of non-discrete examples of locally compact C*-simple groups and Raum showed C*-simplicity of the relative profinite completions of the Baumslag-Solitar groups by using Suzuki's results. We extend this result to some fundamental groups of graphs of groups called generalized Baumslag-Solitar groups. In this article, we focus on some sufficient condition to show that these locally compact groups are C*-simple and that KMS-weights of these reduced group C*-algebras are unique. This condition is an analogue of the Powers averaging property of discrete groups and holds for several currently known constructions of non-discrete C*-simple groups.