Rapidly decreasing functions in reduced $C^ *$-algebras of groups

Abstract:

Let $\Gamma$ be a group. We associate to any length-function $L$ on $\Gamma$ the space $H_L^\infty (\Gamma )$ of rapidly decreasing functions on $\Gamma$ (with respect to $L$), which coincides with the space of smooth functions on the $k$-dimensional torus when $\Gamma = {{\bf {Z}}^k}$. We say that $\Gamma$ has property (RD) if there exists a length-function $L$ on $\Gamma$ such that $H_L^\infty (\Gamma )$ is contained in the reduced ${C^*}$-algebra $C_r^*(\Gamma )$ of $\Gamma$. 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.