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Some results regarding the ideal structure of C*-algebras of étale groupoids

Kevin Aguyar Brix, Toke Meier Carlsen, Aidan Sims

Abstract

We prove a sandwiching lemma for inner-exact locally compact Hausdorff étale groupoids. Our lemma says that every ideal of the reduced $C^*$-algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced $C^*$-algebra, and triples consisting of two nested open invariant sets and an ideal in the $C^*$-algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced $C^*$-algebras of inner-exact locally compact Hausdorff étale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.

Some results regarding the ideal structure of C*-algebras of étale groupoids

Abstract

We prove a sandwiching lemma for inner-exact locally compact Hausdorff étale groupoids. Our lemma says that every ideal of the reduced -algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced -algebra, and triples consisting of two nested open invariant sets and an ideal in the -algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced -algebras of inner-exact locally compact Hausdorff étale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.
Paper Structure (9 sections, 16 theorems, 33 equations)

This paper contains 9 sections, 16 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hausdorff étale groupoids
  4. Reduced groupoid C*-algebra
  5. A sandwiching lemma for Hausdorff étale groupoids
  6. Effectiveness at a unit and the obstruction ideal
  7. Examples
  8. Groupoids from local homeomorphisms
  9. Partial actions

Key Result

Lemma 2.1

Let $G$ be a locally compact Hausdorff étale groupoid. Suppose that $I$ is an ideal of $C^*_r(G)$. Then $\mathop{\mathrm{supp}}\nolimits(I)$ is invariant under multiplication and inversion in $G$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4: The sandwiching lemma
  • proof
  • Remark 3.5
  • ...and 35 more