The representation theory of C*-algebras associated to groupoids

Abstract

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system \lambda. The twisted groupoid C*-algebra C*(E;G,\lambda) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,\lambda) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, \lambda) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, \lambda) has bounded trace if and only if G is integrable and that C*(E;G, \lambda) is a Fell algebra if and only if G is Cartan. Let \G be a second-countable, locally compact, Hausdorff groupoid with Haar system \lambda and continuously varying, abelian isotropy groups. Let A be the isotropy groupoid and R := \G/A. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(\G, \lambda) has bounded trace if and only if R is integrable and that C*(\G, \lambda) is a Fell algebra if and only if R is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.