Cartan Subalgebras in C*Algebras of Haus dorff étale Groupoids
 Citation data:

Integral Equations and Operator Theory, ISSN: 0378620X, Vol: 85, Issue: 1, Page: 109126
 Publication Year:
 2016

 EBSCO 5

 EBSCO 1

 CrossRef 6
 Scopus 6
 Repository URL:
 http://ro.uow.edu.au/eispapers/5449
 DOI:
 10.1007/s0002001622852
 Author(s):
 Publisher(s):
 Tags:
 Mathematics; tale; cartan; groupoids; subalgebras; c; algebras; hausdorff; Engineering; Science and Technology Studies
article description
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 kgraph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.