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

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 bijectio...

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Bibliographic Details
Published in:arXiv.org 2024-01
Main Authors: Brix, Kevin Aguyar, Toke Meier Carlsen, Sims, Aidan
Format: Article
Language:English
Subjects:
Algebra
Invariants
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2211.06126