RIS ID
106441
Abstract
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.
Grant Number
ARC/DP150101595
Publication Details
Brown, J. H., Nagy, G., Reznikoff, S., Sims, A. & Williams, D. P. (2016). Cartan subalgebras in C*-algebras of Hausdorff étale groupoids. Integral Equations and Operator Theory, 85 (1), 109-126.