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Consider an exact action of discrete group $G$ on a separable $C^*$-algebra$A$. It is shown that the reduced crossed product $A\rtimes_{\sigma, \lambda}G$ is strongly purely infinite - provided that the action of $G$ on anyquotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wiseproperly 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 strongpure infiniteness of reduced crossed products of $C^*$-algebras $A$ that arenot $G$-simple. In the case $A=\mathrm{C}_0(X)$ the notion of a $G$-separatingaction corresponds to the property that two compact sets $C_1$ and $C_2$, thatare contained in open subsets $C_j\subseteq U_j \subseteq X$, can be mapped byelements of $g_j\in G$ onto disjoint sets $\sigma_{g_j}(C_j)\subseteq U_j$, butwe do not require that $\sigma_{g_j}(U_j)\subseteq U_j$. A generalization ofstrong boundary actions on compact spaces to non-unital and non-commutative$C^*$-algebras $A$ (cf. Definition 6.1) is also introduced. It is stronger thanthe notion of $G$-separating actions by Proposition 6.6, because $G$-separationdoes not imply $G$-simplicity and there are examples of $G$-separating actionswith reduced crossed products that are stably projection-less and non-simple.

Strong pure infiniteness of crossed products