C*-dynamical rapid decay

Abstract

Some well known results by Haagerup, Jolissaint and de la Harpe may be extended to the setting of a reduced crossed product of a C*-algebra A by a discrete group $G.$ We show that for many discrete groups, which include Gromov's hyperbolic groups and finitely generated discrete groups of polynomial growth, an inequality of the form $$\Vert X\Vert \le C\sqrt{\sum _{g\in G}(1+|g|{)}^4\Vert X_g{\Vert }^2}$$ holds for any finitely supported operator $X$ in the reduced crossed product.

Changes from 1. to 2. submission: Results on "mixed rapid decay" deleted due to an error in the proof. A new reference to de la Harpe's work has generalized and extended the validity of the remaining results