On the structure of certain \(C^*\)-algebras associated to lattices of \(\text{PSL}_2(\mathbb{R})\) (Q1863981)

Consider the Lie group \(G=PSL_2(\mathbb R)\) and let \(\Gamma\) be a lattice in \(G.\) Let \(\pi\) and \(\rho\) be irreducible unitary representations of \(G.\) Assume that \(\pi\) and \(\rho\) are not equivalent and that they do not belong to the discrete series in \(G.\) It was shown by \textit{M. Cowling} and \textit{T. Steger} [J. Reine Angew. Math. 420, 85-98 (1991; Zbl 0760.22014)] that the restrictions \(\pi|_{\Gamma}\) and \(\rho|_{\Gamma}\) are irreducible representations of \(\Gamma\) and that they are not equivalent. Let \(\pi\) be an irreducible unitary representation of \(G,\) different from the trivial representation. Let \(C^*_{\pi}(\Gamma)\) be the \(C^*\)-algebra generated by all the unitary operators \(\pi(\gamma), \gamma\in \Gamma.\) It is natural to ask: what is the structure of \(C^*_{\pi}(\Gamma)?\) It was shown by \textit{M. Bekka} and \textit{P. de la Harpe} [Bull. Soc. Math. France 122, 333-342 (1994; Zbl 0824.22011)] that the reduced \(C^*\)-algebra \(C^*_{r}(\Gamma)\) of \(\Gamma\) is always a quotient of \(C^*_{\pi}(\Gamma).\) So, if \(\pi\) belongs to the principal series, then \(C^*_{\pi}(\Gamma)\) is isomorphic to \(C^*_{r}(\Gamma).\) Assume now that \(\pi\) belongs to the complementary series of \(G.\) In the paper under review, the author gives a complete description of \(C^*_{\pi}(\Gamma).\) In details: he shows that \(C^*_{\pi}(\Gamma)\) is a split extension of \(C^*_{r}(\Gamma)\) by the algebra of all compact operators on a separable Hilbert space. Moreover, he shows that \(C^*_{\pi}(\Gamma)\) and \(C^*_{\rho}(\Gamma)\) are isomorphic for all complementary series representations \(\pi\) and \(\rho.\) It is worth mentioning that, nevertheless, \(\pi|_{\Gamma}\) and \(\rho|_{\Gamma}\) are not weakly equivalent if \(\pi\) and \(\rho\) are not equivalent (in fact, neither of these representations is weakly contained in the other).