We prove that the full C * -algebra of a second-countable, Hausdorff,étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex * -algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.

Abstract. We present a uniqueness theorem for k-graph C*-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C*-algebra, it is sufficient that the representation be injective on a distinguished abelian C*-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C*-algebra, each of which is the unique extension of a state on the distinguished abelian C*-subalgebra.

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.

Abstract. Let G be a Hausdorff,étale groupoid that is minimal and topologically principal. We show that C * r (G) is purely infinite simple if and only if all the nonzero positive elements of C 0 (G (0) ) are infinite in C * r (G). If G is a Hausdorff, ample groupoid, then we show that C * r (G) is purely infinite simple if and only if every nonzero projection in C 0 (G (0) ) is infinite in C * r (G). We then show how this result applies to k-graph C * -algebras. Finally, we investigate strongly purely infinite groupoid C * -algebras.

We study graph C *-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph C *-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding two-sided subshifts are conjugate.