Stone duality and quasi-orbit spaces for generalised C*-inclusions

Abstract

Let $A$ and $B$ be $C^{\ast }$-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 $C^{\ast }$-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.

Locale-theoretical notation changed, an error in a previous version of Lemma 3.10 removed, and a number of small changes implemented

References in corpus (5)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Fell bundles over inverse semigroups and twisted etale groupoids
• Equivalence of Fell bundles over groups
• Morita enveloping Fell bundles
• Topological freeness for $C^{\ast }$-correspondences

Cited by in corpus (4)

• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Topologically free actions and ideals in twisted Banach algebra crossed products
• Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
• Maximal ideals of reduced group C*-algebras and Thompson's groups