Stone duality and quasi‐orbit spaces for generalised C∗‐inclusions

Let $A$ and $B$ be $C^*$-algebras with $A\subseteq M(B)$. Exploiting Stone duality and a Galois connection between restriction and induction for ideals in $A$ and $B$, we identify conditions that allow to define a quasi-orbit space and a quasi-orbit map for $A\subseteq M(B)$. These objects generalise classical notions for group actions. We characterise when the quasi-orbit space is an open quotient of the primitive ideal space of $A$ and when the quasi-orbit map is open and surjective. We discuss applications of these results to cross section $C^*$-algebras of Fell bundles over locally compact groups, regular $C^*$-inclusions, tensor products, relative Cuntz-Pimsner algebras, and crossed products for actions of locally compact Hausdorff groupoids and quantum groups.