Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

Abstract

We study simplicity and pure infiniteness criteria for C ˚algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of Exel and Pitts, we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced C ˚-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic C ˚-inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

Left regular representations of Garside categories I. C*-algebras and groupoids

“…As explained in [37, Section 2.1], we call the transformation groupoid and not the groupoid of germs (as in for instance [25]) because in other contexts, the groupoid of germs denotes the quotient of a groupoid by the interior of its isotropy subgroupoid (see for instance [63]).…”

Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids

“…It then follows from a computation that {ξ i,n } witnesses the approximation property for A. Likewise, by [24,Example 4.7], if G is non-amenable then C * max (A) ∼ = C * red (A). By Theorem 6.2 below it follows that A does not have the approximation property.…”

“…Consider the Banach spaces A g := C[0, 1] whenever g ∈ G and A 0 := C(0, 1]. It is shown in [24,Example 4.7] that A = (A s ) s∈S forms a Fell bundle over S. Moreover, it is also mentioned that both C * max (A) and C * red (A) are C[0, 1]-algebras with trivial fibers at x ∈ (0, 1]. Nevertheless, their fiber at 0 may differ.…”

“…For the purposes of this paper, however, the construction of C * ess (A) is irrelevant, albeit for the fact that it is, canonically, a quotient of C * red (A) in the following way. Theorem 6.8 ([24,Theorem 4.11]). Given a Fell bundle A = (A s ) s∈S over S, the identity map on C c (A) induces canonical quotient maps…”

“…Example 6.10. Let G be a group, and let S and A = (A s ) s∈S be as in Example 4.9 (see also [24,Example 4.7]). By the discussion in Example 4.9, if G is non-amenable then A does not have the approximation property, and the left regular representation Λ : C * max (A) → C * red (A) is not injective.…”