The representation theory of C*-algebras associated to groupoids

TL;DR

This paper characterizes the representation-theoretic properties of C*-algebras associated with groupoids, linking topological conditions of orbit spaces and integrability to algebraic properties like liminal, postliminal, and Fell algebra status.

Contribution

It provides new criteria connecting the topological structure of groupoid orbit spaces and isotropy groups to the properties of their associated C*-algebras, extending previous understanding.

Findings

C*-algebra is postliminal iff the orbit space is T0.

C*-algebra is liminal iff the orbit space is T1.

C*-algebra has bounded trace iff the groupoid is integrable.

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.