Continuous-trace groupoid 𝐶*-algebras. III

Abstract

Abstract. Suppose that G is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid C * -algebra C * (G, λ) has continuous trace if and only if there is a Haar system for the isotropy groupoid A and the action of the quotient groupoid G/A is proper on the unit space of G.

Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

“…It will have Exel's strong property if and only if the group G is amenable. [17,23] (assuming that our groupoid is second countable), we are in position to give an easy to use sufficient condition for a groupoid to be Fredholm. The above proposition can be generalized to groupoids given by a filtration.…”

Function Algebras on <em>r</em>-Discrete Abelian Groupoids

“…In [11], the authors show that there is generalized conditional expectation C * (Γ) → C * (Γ(x)), if we add these generalized conditional expectations, we get a generalized conditional expectation C * (Γ) → C * (Γ(X)). In [4, Theorem 3.4], the following corollary is proved for the r-discrete case.…”

“…The subgroupoids Γ(X) and R(Γ) are measured groupoids. Note that for a measured groupoid, the continuity of the Haar systems β and α holds only in particular cases (for example see [11] and [7]). In these cases, the above groupoids are locally compact groupoids.…”

Pushouts of extensions of groupoids by bundles of abelian groups

“…Specifically, the action of G on A induces a natural action of G on (regarded as a space). We constructed a T-groupoid Σ of the form A × T Σ G. The chief motivation for this article is the observation that the T-groupoid Σ above-which was based on the construction of [MRW96,Proposition 4.3]-is derived from a natural and functorial "pushout" construction based on the second author's work in [Kum88] forétale groupoids (there called "sheaf groupoids"). Specifically, suppose we are given an extension as in ( †), an abelian group bundle B admitting a G-action, and an equivariant groupoid homomorphism f : A → B.…”