Cartan subalgebras in C*-algebras of Hausdorff etale groupoids

TL;DR

This paper investigates the structure of Cartan subalgebras within the reduced C*-algebras of Hausdorff étale groupoids, establishing conditions for their existence, maximality, and properties related to isotropy and representations.

Contribution

It characterizes when the interior of the isotropy yields a Cartan subalgebra and identifies classes of groupoids where this subalgebra is maximal abelian but not necessarily Cartan.

Findings

Pure states with unique extension are dense in the subalgebra.

Injective representations on the subalgebra are faithful.

When isotropy is abelian and closed, the subalgebra is Cartan.

Abstract

The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff \'etale 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.