Simplicity of twisted C*-algebras of Deaconu--Renault groupoids

Abstract

We consider Deaconu-Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2-cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.

Conjugacy of local homeomorphisms via groupoids and C*-algebras

“…A system consisting of a locally compact Hausdorff space X together with a local homeomorphism σ X between open subsets of X is called a Deaconu-Renault system (see, for example, [ABS,CRST,D95,Re00]). Examples of Deaconu-Renault systems include self-covering maps [D95,EV06], one-sided shifts of finite type [Ki98,LM95,Wi73], the boundary-path space of a directed graph together with the shift map [BCW17,We14], and, more generally, the boundary-path space of a topological graph together with the shift map [KL17], the one-sided edge shift space of an ultragraph together with the restriction of the shift map to points with non-zero length [GR19], the full one-sided shift over an infinite alphabet together with the restriction of the shift map to points with non-zero length [OMW14], the cover of a one-sided shift space constructed in [BC20b], and, more generally, the canonical local homeomorphism extension of a locally injective map constructed in [Th11].…”