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

TL;DR

This paper develops a general framework for quasi-orbit spaces and maps in $C^*$-algebra inclusions, extending classical group action notions using duality and locale theory, with applications to various algebraic constructions.

Contribution

It introduces a unified approach to quasi-orbit spaces for $C^*$-inclusions, generalizing classical concepts and characterizing their properties through duality and locale theory.

Findings

Conditions for defining quasi-orbit spaces and maps.

Characterization of when the quasi-orbit space is an open quotient.

Applications to Fell bundles, Cuntz--Pimsner algebras, and crossed products.

Abstract

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.