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Journal of Operator Theory

Volume 57, Issue 2, Spring 2007  pp. 251-266.

Classifying the types of principal Groupoid $C^*$-algebras

Authors Lisa Orloff Clark
Author institution: Department of Mathematical Sciences, Susquehanna University, Selinsgrove, PA 17870, USA

Summary:  Suppose $G$ is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let $G^0/G$ denote the orbit space of $G$ and $C^*(G)$ denote the groupoid $C^*$-algebra. Suppose that $G$ is a principal groupoid. We show that $C^*(G)$ is CCR if and only if $G^0/G$ is a $T_1$ topological space, and that $C^*(G)$ is GCR if and only if $G^0/G$ 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.

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