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

Becky Armstrong, Nathan Brownlowe, Aidan Sims

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.

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.

Paper Structure

This paper contains 15 sections, 38 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Group 2-cocycles and bicharacters
  4. Hausdorff etale groupoids
  5. Cohomology of groupoids
  6. Twisted groupoid C*-algebras
  7. Deaconu--Renault groupoids
  8. The interior of the isotropy of a Deaconu--Renault groupoid
  9. Cohomology of Deaconu--Renault groupoids
  10. Realising C*(IT,sigma) as an induced algebra
  11. Simplicity of twisted C*-algebras of Deaconu-Renault groupoids
  12. An application to some crossed products by the integers
  13. Crossed products as twisted groupoid C*-algebras
  14. An application to crossed products
  15. Realising twisted group C*-algebras as induced algebras

Key Result

Lemma \oldthetheorem

Let $(X,T)$ be a rank-$k$ Deaconu--Renault system. The map $c\colon (x,n,y) \mapsto n$ is a continuous $\mathbb{Z}^k$-valued $1$-cocycle on $\mathcal{G}_T$, and for each $x \in X$, the restriction of $c$ to $(\mathcal{G}_T)_x^x$ is injective.

Theorems & Definitions (91)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • ...and 81 more