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

de la Harpe, Pierre

C. R. Acad. Sci., Paris, Sér. I 307, No. 14, 771-774 (1988).

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.

MSC:

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

Keywords:

Operator algebras defined by Gromov’s hyperbolic groups; reduced C *- algebra; von Neumann; holomorphic calculus; torsion free; simple C *- algebra with unique trace; full factor