\(C^\ast \)-simplicity of relative profinite completions of generalized Baumslag-Solitar groups (Q6618962)

The reduced \(C^*\)-algebra \(C^*_r (G)\) of a locally compact group \(G\) is the norm closure of the algebra of operators on \(L^2(G)\) generated by the left regular \(*\)-representation \(\lambda\) of \(C_c(G)\). The group \(G\) is said to be \(C^*\)-simple if \(C^*_r (G)\) is simple. Since the proof of the \(C^*\)-simplicity of the free group of rank two by \textit{R. T. Powers} [Duke Math. J. 42, 151--156 (1975; Zbl 0342.46046)], the study of the class of \(C^*\)-simple groups has begun. In the case of discrete groups, there are several other results concerning \(C^*\)-simplicity. For example it was proved that if \(G\) is a \(C^*\)-simple group, then its amenable radical is trivial. It is also known that a discrete group \(G\) is \(C^*\)-simple if and only if for each \(a\in C^*_r (G)\) with canonical trace \(\tau\), \(\tau(a)\) is approximated by elements in the convex hull of the set \(\{\lambda_s a \lambda_s^*\in C^*_r (G):\ s\in G\}\). Suzuki constructed the first example of a non-discrete \(C^*\)-simple group. \textit{S. Raum} proved in [J. Reine Angew. Math. 748, 173--205 (2019; Zbl 1419.46035)] that for \(|n|,|m|>1\) the relative profinite completion of the celebrated Baumslag-Solitar group is \(C^*\)-simple. In the present paper as an extension of this result, the author gives a proof to show the \(C^*\)-simplicity of generalized Baumslag-Solitar groups which are some fundamental groups of graphs of groups.