Some groups whose reduced \(C^*\)-algebras have stable rank one
The authors prove that the reduced group \(C^\ast\)-algebra \(C^\ast_{\text{red}}(\Gamma)\) of a discrete group \(\Gamma\) has stable rank one if \(\Gamma\) has (1) the free semigroup property and (2) the \(\ell^2\)-spectral radius property. This means that for any finite subset \(F\subset\Gamma\) there is \(\gamma\in\Gamma\) such that (1) the semigroup generated by \(\gamma F\) is free over \(F\) and that (2) for any \(a\in\text{span } \gamma F\) the spectral radius can be computed from the trace norm of \(\Gamma\). In particular, this holds for amalgamated free products \(\Gamma=G_1\ast_H G_2\) with certain finite \(H\) and for hyperbolic groups which are either torsion-free and non-elementary or represented as a cocompact lattice in a real, noncompact, simple, connected Lie group of real rank 1 and trivial center. Finally, they also discuss the real rank and formulate some open problems.
scientific article; zbMATH DE number 1344452