Cartan subalgebras in C*-algebras of Hausdorff étale groupoids

Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, Dana P. Williams

Published 2015-03-11Version 1

The reduced C*-algebra of the interior of the isotropy in any Hausdorff \'etale groupoid G embeds as a C*-subalgebra of the reduced C*-algebra of G. We prove that any representation of the reduced algebra of G that is injective on this subalgebra is faithful. We also show that restriction of functions extends to a faithful conditional expectation of the reduced C*-algebra of G onto the embedded subalgebra, and the set of pure states of the subalgebra with unique extension to the larger C*-algebra is dense. If the interior of the isotropy is abelian, then the embedded subalgebra is a Cartan subalgebra in the sense of Renault.