  Let $A$ and $B$ be $C^*$-algebras with $A\subseteq M(B)$. Exploiting the
duality between sober spaces and spatial locales, and the adjunction 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
apply 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.
