Strong pure infiniteness of crossed products
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 G-invariant closed ideal I 6= 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 = C0(X) the notion of a G-separating 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  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.
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