Some results regarding the ideal structure of C*-algebras of \'etale groupoids

TL;DR

This paper establishes a new structural understanding of ideals in the reduced C*-algebras of inner-exact étale groupoids, linking them to open invariant subsets and introducing a generalized topological freeness condition.

Contribution

It proves a sandwiching lemma for ideals, characterizes all ideals via open invariant sets and subquotients, and introduces a generalized topological freeness condition for groupoids.

Findings

Every ideal is between two open invariant subset ideals.

A bijection exists between ideals and triples of nested open sets and subquotient ideals.

Inner-exact groupoids satisfying the generalized freeness condition admit an obstruction ideal.

Abstract

We prove a sandwiching lemma for inner-exact locally compact Hausdorff \'etale 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 \'etale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.