CCR and GCR Groupoid C*-algebras

Abstract

Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $\go/G$ denote the orbit space of $G$ and $C^{\ast }(G)$ denote the groupoid $C^{\ast }$-algebra. Suppose that the isotropy groups of $G$ are amenable. We show that $C^{\ast }(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space and all of the isotropy groups are CCR. We also show that $C^{\ast }(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space and all of the isotropy groups are GCR.