Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint. (Hyperbolic groups, operator algebras and Jolissaint’s theorem). (French. English summary) Zbl 0653.46059

Summary: Operator algebras defined by Gromov’s hyperbolic groups have interesting properties. Let \(\Gamma\) be such a group, let \(C\) \(*_ r(\Gamma)\) be its reduced C *-algebra and let W *(\(\Gamma)\) be its von Neumann algebra. Our main observation is that the following theorem, due to Jolissaint, holds for \(C\) \(*_ r(\Gamma):\) rapidly decreasing functions constitute a subalgebra which is stable by holomorphic calculus. If \(\Gamma\) is torsion free, we observe moreover that \(C\) \(*_ r(\Gamma)\) is a simple C *-algebra with unique trace and that W *(\(\Gamma)\) is a full factor.


46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
46L35 Classifications of \(C^*\)-algebras
46H30 Functional calculus in topological algebras