\(C^\ast \)-simplicity of relative profinite completions of generalized Baumslag-Solitar groups. (English) Zbl 07926395
Summary: Suzuki recently gave constructions of non-discrete examples of locally compact \(C^\ast \)-simple groups and Raum showed \(C^\ast \)-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 \(C^\ast \)-simple and that KMS-weights of these reduced group \(C^\ast \)-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 \(C^\ast \)-simple groups.
MSC:
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 46L05 | General theory of \(C^*\)-algebras |
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