Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

Abstract

Abstract. The reduced C * -algebra of the interior of the isotropy in any Hausdorf 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.

Simplicity of algebras associated to non-Hausdorff groupoids

“…In our proof, we will invoke Theorem 3.2 of [3] which we restate for convenience. (1) every ϕ ∈ S has a unique extension to a stateφ of A; and (2) the direct sum ⊕ ϕ∈S πφ of the GNS representations associated to extensions of the elements of S to A is faithful on A.…”

Reduced C*-algebras of Fell bundles over inverse semigroups

“…Thus the criterion for E(A r S) ⊆ A in Proposition 6.3 is equivalent to D 1,t being relatively closed in D t for each t ∈ S. The open subsets D t ⊆ (Prim(A) S) 1 form an open covering, and D 1,t = D t ∩ Prim(A). Hence D 1,t is relatively closed in D t for each t ∈ S if and only if the subset of units Prim(A) is closed in (Prim(A) S) 1 .The theorem above is related to[3, Corollary 4.4].The existence of a conditional expectation A r S → A should be viewed as an analogue for inverse semigroup crossed products of Hausdorffness for groupoid crossed products. By Lemma 5.2, a groupoid with Hausdorff object space has Hausdorff arrow space if and only if the set of units is closed.…”

Condition (K) for inverse semigroups and the ideal structure of their C⁎-algebras

“…The theory of graph C * -algebras has inspired a number of generalizations, and consequently it has provided motivation for a large body of research in C * -algebras in recent years. In some ways this paper is a follow up to [11], another paper inspired by graph C * -algebras results, in which two uniqueness theorems for C * -algebras of inverse semigroups were proved using the generalized uniqueness theorem forétale groupoids from [3]. If one is to follow the program laid out in the graph C * -algebra literature, the natural next step is to describe the ideal structure of the tight C * -algebra of an inverse semigroup.…”