Strong pure infiniteness of crossed products

Consider an exact action of a discrete group $G$ on a separable C*-algebra $A$ . It is shown that the reduced crossed product $A\rtimes _{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706}}G$ is strongly purely infinite—provided that the action of $G$ on any quotient $A/I$ by a $G$ -invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$ -separating (cf. Definition 5.1). This is the first non-trivial sufficient general criterion for strong pure infiniteness of reduced crossed products of C*-algebras $A$ that are not $G$ -simple. In the case $A=\text{C}_{0}(X)$ , 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 $C_{j}\subseteq U_{j}\subseteq X$ , can be mapped by elements $g_{1},g_{2}$ of $G$ onto disjoint sets $\unicode[STIX]{x1D70E}_{g_{j}}(C_{j})\subseteq U_{j}$ , but satisfy not necessarily the contraction property $\unicode[STIX]{x1D70E}_{g_{j}}(U_{j})\subseteq \overline{U_{j}}$ . A generalization of strong boundary actions on compact spaces to non-unital and non-commutative C*-algebras $A$ (cf. Definition 7.1) is also introduced. It is stronger than the notion of $G$ -separating actions by Proposition 7.6, 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.