Article ID Journal Published Year Pages File Type
8905024 Advances in Mathematics 2018 63 Pages PDF
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 Max A 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 Max A 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.
Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
Authors
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