Fourier coefficients and rapid decay in reduced groupoid C*-algebras

Abstract

Let $\Sigma \to G$ be a twist over a locally compact Hausdorff étale groupoid $G$. Given $f$ in the reduced C${}^{\ast }$-algebra $C_r^{\ast }(\Sigma ;G)$ with open support $U\subseteq G$ we ask when $f$ lies in the closure of the compactly supported sections on $U$. Suppose $G$ satisfies the rapid decay property with respect to a length function $L$. We give a positive answer to our question in two instances: when $L$ is conditionally negative-definite, and when $L$ is the square-root of a locally negative type function on $G$.

15 pages