Rapidly decreasing functions in reduced 𝐶*-algebras of groups

Source abstract

Let Γ \Gamma be a group. We associate to any length-function L L on Γ \Gamma the space H L ∞ ( Γ ) H_L^\infty (\Gamma ) of rapidly decreasing functions on Γ \Gamma (with respect to L L ), which coincides with the space of smooth functions on the k k -dimensional torus when Γ = Z k \Gamma = {{\bf {Z}}^k} . We say that Γ \Gamma has property (RD) if there exists a length-function L L on Γ \Gamma such that H L ∞ ( Γ ) H_L^\infty (\Gamma ) is contained in the reduced C ∗ {C^*} -algebra C r ∗ ( Γ ) C_r^*(\Gamma ) of Γ \Gamma . We study the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products. Finally, we show that the following groups have property (RD): (1) Finitely generated groups of polynomial growth; (2) Discrete cocompact subgroups of the group of all isometries of any hyperbolic space.