Quantales and Fell bundles

Abstract

We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle $\pi :E\to G$ on an étale groupoid $G$ with $G_0$ locally compact Hausdorff, equipped with a suitable completion C*-algebra $A$ of its convolution algebra, we obtain a map of involutive quantales $p:\mathrm{Max}A\to \Omega (G)$, where $\mathrm{Max}A$ consists of the closed linear subspaces of $A$ and $\Omega (G)$ is the topology of $G$. We study various properties of $p$ which mimick, to various degrees, those of open maps of topological spaces. These are closely related to properties of $G$, $\pi$, and $A$, such as $G$ being Hausdorff, principal, or topological principal, or $\pi$ being a line bundle. Under suitable conditions, which include $G$ being Hausdorff, but without requiring saturation of the Fell bundle, $A$ is an algebra of sections of the bundle if and only if it is the reduced C*-algebra $C_r^{\ast }(G,E)$. We also prove that $\mathrm{Max}A$ is stably Gelfand. This implies the existence of a pseudogroup ${\mathcal{I}}_B$ and of an étale groupoid $\mathfrak{B}$ associated canonically to any sub-C*-algebra $B\subset A$. We study a correspondence between Fell bundles and sub-C*-algebras based on these constructions, and compare it to the construction of Weyl groupoids from Cartan subalgebras.

Version 2 contains a thorough revision of the paper. It fixes some mistakes and presentation issues, and includes new material related to principal groupoids, topologically principal groupoids, and groupoids associated to sub-C*-algebras. Version 3 is the final journal version (modulo possible typos or reference updates)

References in corpus (11)

• Etale groupoids and their quantales
• On quantales that classify C*-algebras
• Inverse semigroup actions on groupoids
• Fell bundles over inverse semigroups and twisted etale groupoids
• Noncommutative Cartan sub-algebras of C*-algebras
• Functoriality of groupoid quantales. I
• Open quotients of trivial vector bundles
• On quantales and spectra of C*-algebras
• The many groupoids of a stably Gelfand quantale
• Linear structures on locales
• Open maps of involutive quantales

Cited by in corpus (5)

• An Algebraic Approach to the Weyl Groupoid
• The many groupoids of a stably Gelfand quantale
• An abstract theory of physical measurements
• Open maps of involutive quantales
• On the geometry of physical measurements: topological and algebraic aspects