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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