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 $C^*$-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.