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

TL;DR

This paper investigates conditions under which elements of reduced groupoid C*-algebras with open support can be approximated by compactly supported sections, focusing on groupoids with the rapid decay property and specific length functions.

Contribution

It establishes criteria for the approximation of algebra elements by compactly supported sections in reduced groupoid C*-algebras under rapid decay conditions with particular length functions.

Findings

Positive results for approximation when length function is conditionally negative-definite.

Positive results when length function is the square-root of a locally negative type function.

Provides new insights into decay properties in the context of groupoid C*-algebras.

Abstract

Let $\Sigma \rightarrow G$ be a twist over a locally compact Hausdorff \'{e}tale groupoid $G$. Given $f$ in the reduced C$^*$-algebra $C_r^*(\Sigma;G)$ 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 $G$.