Quantales and Fell bundles

Abstract

We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle π:E→G on an étale groupoid G with G0 locally compact Hausdorff, equipped with a suitable completion C*-algebra A of its convolution algebra, we obtain a map of involutive quantales p:MaxA→Ω(G), where MaxA consists of the closed linear subspaces of A and Ω(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, π, and A, such as G being Hausdorff, principal, or topological principal, or π 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 Cr⁎(G,E). We also prove that MaxA is stably Gelfand. This implies the existence of a pseudogroup IB and of an étale groupoid B associated canonically to any sub-C*-algebra B⊂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.