C*-actions of r-discrete groupoids and inverse semigroups

Abstract

Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras are closely related when the groupoid is r-discrete.

Inverse semigroup actions on groupoids

“…A subsemigroup S ⊆ Bis(G) with the properties required in Corollary 3.19 is called wide. Corollary 3.19 explains why they appear so frequently (see, for instance, [3,12,28]). [12,Proposition 5.4] already shows that Z ⋊ S = G if S is wide, but we have not seen the converse statement yet.…”

“…In the end, we want an action of the groupoid G itself, not of the inverse semigroup Bis(G). For actions by automorphisms, Sieben and Quigg [28] characterise which actions of Bis(G) come from actions of G. We extend this characterisation to Fell bundles: a Fell bundle over Bis(G) comes from a Fell bundle over G if and only if the restriction of the action to idempotents in Bis(G) commutes with suprema of arbitrarily large subsets. This criterion only works for Bis(G) itself.…”

Simplicity of algebras associated to étale groupoids

“…Though some parts of what we have done can be found in the literature, we have taken pains to make our results self-contained and to take the most elementary path possible. There are many classes of C * -algebras withétale groupoid models (see for example [8,11,13,14,16,19,26,28,31,36]), so we expect that our results will find numerous applications.…”

An elementary Green imprimitivity theorem for inverse semigroups

“…7.2]. To meet exactly the assumptions in [15], switch to the carrier algebraà = p(A) for p = e∈H (0) e of A, which does not change the crossed product, that is,à ⋊H ′ = A ⋊H ′ .…”