Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - July 2012 - Volume 153, Issue 01  

Research Article

The representation theory of C*-algebras associated to groupoids

LISA ORLOFF CLARKa1 and ASTRID AN HUEFa1

a1 Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand. e-mail: lclark@maths.otago.ac.nz, astrid@maths.otago.ac.nz

Abstract

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.

Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

(Received July 16 2010)

(Revised October 26 2011)

(Online publication February 27 2012)