Some results regarding the ideal structure of C∗$C^*$‐algebras of étale groupoids

Abstract

AbstractWe 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.