${\mathrm{C}}^{\ast }$-Simplicity of Relative Profinite Completions of Generalized Baumslag–Solitar Groups

Abstract

Suzuki recently gave constructions of non-discrete examples of locally compact ${\mathrm{C}}^{\ast }$-simple groups and Raum showed ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-simple and that KMS-weights of these reduced group ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-simple groups.