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 \'etale 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 $C_r^*(G,E)$. We also prove that $\mathrm{Max}\ A$ is stably Gelfand. This implies the existence of a pseudogroup $\mathcal{I}_B$ and of an \'etale 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.Comment: 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

An Abstract Theory of Physical Measurements

“…Let A be a C*-algebra, and let Max A be Mulvey's involutive quantale of closed linear subspaces of A [24,25], whose involution is computed pointwise from the involution of A and whose product is defined to be the closure of the linear span of the pointwise product: This is a stably Gelfand quantale [30], and it is a sober space if we equip it with the lower Vietoris topology [33], whose open sets are the unions of finite intersections of the form Ũ1 ∩ … ∩ Ũk , where for each open set U of A we have It can be proved that the algebraic operations of Max A are continuous [32], so Max A is a measurement space.…”

An Algebraic Approach to the Weyl Groupoid

“…Recently there has been a push to extend the Weyl groupoid construction in various directions, e.g. in [CRST17], [Res18], [EP19] and [KM19], but even in these generalisations, some restrictions on the isotropy have remained.…”