We study conditions that will ensure that a crossed product of a C-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C-algebra is a Kirchberg algebra in the UCT class. Abstract. We study conditions that will ensure that a crossed product of a C * -algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C * -algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C * -algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C * -algebra is a Kirchberg algebra in the UCT class.

Let (A, G, α) be a partial dynamical system. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A ⋊α,r G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition-automatic when A is abelian-we show that A ⋊α,r G is purely infinite if and only if the positive nonzero elements in A are properly infinite in A ⋊α,r G. As an application we verify pure infiniteness of various partial crossed products, including realisations of the Cuntz algebras On, OA, O N , and O Z as partial crossed products.

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.

Abstract. Consider an exact action of discrete group G on a separable C *-algebra A. It is shown that the reduced crossed product A ⋊ σ,λ G is strongly purely infinite -provided that the action of G on any quotient A/I by a G-invariant closed ideal I = A is element-wise properly outer and that the action of G on A is G-separating (cf. Definition 4.1). This is the first non-trivial sufficient criterion for strong pure infiniteness of reduced crossed products of C *-algebras A that are not G-simple. In the case A = C 0 (X) the notion of a G-separating action corresponds to the property that two compact sets C 1 and C 2 , that are contained in open subsets C j ⊆ U j ⊆ X, can be mapped by elements of g j ∈ G onto disjoint sets σ gj (C j ) ⊆ U j , but we do not require that σ gj (U j ) ⊆ U j . A generalization of strong boundary actions [18] on compact spaces to non-unital and non-commutative C *-algebras A (cf. Definition 6.1) is also introduced. It is stronger than the notion of G-separating actions by Proposition 6.6, because G-separation does not imply G-simplicity and there are examples of G-separating actions with reduced crossed products that are stably projection-less and non-simple.

It was shown by Rørdam and the second named author that a countable group G admits an action on a compact space such that the crossed product is a Kirchberg algebra if, and only if, G is exact and non-amenable. This construction allows a certain amount of choice. We show that different choices can lead to different algebras, at least with the free group.1991 Mathematics Subject Classification. Primary 46L35, Secondary 46L80, 19K99.