Abstract
We give efficient conditions under which a \mathrm{C}^* -subalgebra A\subseteq B separates ideals in a \mathrm{C}^* -algebra B , and B is purely infinite if every positive element in A is properly infinite in B . We specialise to the case when B is a crossed product for an inverse semigroup action by Hilbert bimodules or a section \mathrm{C}^* -algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided B is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article.
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