Classifying the Type of Principal 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 $\cs(G)$ denote the groupoid $C^{\ast }$-algebra. Suppose that $G$ is a principal groupoid. We show that $C^{\ast }(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space, and that $C^{\ast }(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space. We also show that $C^{\ast }(G)$ is a Fell Algebra if and only if $G$ is a Cartan groupoid.