Abstract

Consider an exact action of discrete group G on a separable C*algebra A. It is shown that the reduced crossed product A ⋊ G is strongly purely infinite – provided that the action of G on any quotient A/I by a Ginvariant closed ideal I 6= A is elementwise properly outer and that the action of G on A is Gseparating (cf. Definition 4.1). This is the first nontrivial sufficient criterion for strong pure infiniteness of reduced crossed products of C*algebras A that are not Gsimple. In the case A = C0(X) the notion of a Gseparating action corresponds to the property that two compact sets C1 and C2, that are contained in open subsets Cj ⊆ Uj ⊆ X, can be mapped by elements of gj ∈ G onto disjoint sets gj (Cj ) ⊆ Uj , but we do not require that gj (Uj) ⊆ Uj . A generalization of strong boundary actions [18] on compact
spaces to nonunital and noncommutative C*algebras A (cf. Definition 6.1) is also introduced. It is stronger than the notion of Gseparating actions by Proposition 6.6, because Gseparation does not imply Gsimplicity and there are examples of Gseparating actions with reduced crossed products that are stably projectionless and nonsimple.