Cartan subalgebras in C-algebras of Hausdorff etale groupoids

The reduced \(C^*\)-algebra of the interior of the isotropy in any Hausdorff étale groupoid \(G\) embeds as a \(C^*\)-subalgebra \(M\) of the reduced \(C^*\)-algebra of \(G\). We prove that the set of pure states of \(M\) with unique extension is dense, and deduce that any representation of the redu...

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Bibliographic Details
Published in:arXiv.org 2016-05
Main Authors: Brown, Jonathan H, Nagy, Gabriel, Reznikoff, Sarah, Sims, Aidan, Williams, Dana P
Format: Article
Language:English
Subjects:
Algebra
Group theory
Isotropy
Online Access:Get full text
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Summary:The reduced \(C^*\)-algebra of the interior of the isotropy in any Hausdorff étale groupoid \(G\) embeds as a \(C^*\)-subalgebra \(M\) of the reduced \(C^*\)-algebra of \(G\). We prove that the set of pure states of \(M\) with unique extension is dense, and deduce that any representation of the reduced \(C^*\)-algebra of \(G\) that is injective on \(M\) is faithful. We prove that there is a conditional expectation from the reduced \(C^*\)-algebra of \(G\) onto \(M\) if and only if the interior of the isotropy in \(G\) is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, \(M\) is a Cartan subalgebra. We prove that for a large class of groupoids \(G\) with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---\(M\) is a maximal abelian subalgebra. In the specific case of \(k\)-graph groupoids, we deduce that \(M\) is always maximal abelian, but show by example that it is not always Cartan.
ISSN:2331-8422
DOI:10.48550/arxiv.1503.03521