Classifying the Type of Principal Groupoid C*-algebras

TL;DR

This paper characterizes when principal groupoid C*-algebras are CCR, GCR, or Fell algebras based on topological properties of the orbit space, linking algebraic types to topology.

Contribution

It establishes precise conditions relating the type of principal groupoid C*-algebras to the topological separation properties of their orbit spaces.

Findings

CCR if and only if orbit space is T1

GCR if and only if orbit space is T0

Fell algebra if and only if groupoid is Cartan

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 $C^*$-algebra. Suppose that $G$ is a principal groupoid. We show that $C^*(G)$ is CCR if and only if $\go/G$ is a $T_1$ topological space, and that $C^*(G)$ is GCR if and only if $\go/G$ is a $T_0$ topological space. We also show that $C^*(G)$ is a Fell Algebra if and only if $G$ is a Cartan groupoid.