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Stone duality and quasi-orbit spaces for generalised \(C^\ast\)-inclusions. (English) Zbl 1461.46070

Summary: Let \(A\) be a \(C^\ast\)-subalgebra of the multiplier algebra \(\mathcal{M}(B)\) of a \(C^\ast\)-algebra \(B\). 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 \(A\subseteq\mathcal{M}(B)\). 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 \(C^\ast\)-algebras of Fell bundles over locally compact groups, regular \(C^\ast\)-inclusions, tensor products, relative Cuntz-Pimsner algebras and crossed products for actions of locally compact Hausdorff groupoids and quantum groups.

MSC:

46L55 Noncommutative dynamical systems
06D22 Frames, locales
20M18 Inverse semigroups
22A22 Topological groupoids (including differentiable and Lie groupoids)
20G42 Quantum groups (quantized function algebras) and their representations
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