Group bundle duality

Abstract

This paper introduces a generalization of Pontryagin duality for locally compact Hausdorff Abelian groups to locally compact Hausdorff Abelian group bundles. First, recall that a group bundle is just a groupoid where the range and source maps coincide. An Abelian group bundle is a bundle where each fibre is an Abelian group. When working with a group bundle G, we will use X to denote the unit space of G and p : G→X to denote the combined range and source maps. Furthermore, we will use Gx to denote the fibre over x. Group bundles, like general groupoids, may not have a Haar system but when they do the Haar system has a special form. If G is a locally compact Hausdorff group bundle with Haar system, denoted by {βx} throughout the paper, then β is Haar measure on the fibre Gx for all x ∈X . At this point, it is convenient to make the standing assumption that all of the locally compact spaces in this paper are Hausdorff. Now suppose G is an Abelian, second countable, locally compact group bundle with Haar system {βx}. Then C∗(G,β) is a separable Abelian C∗algebra and in particular Ĝ=C∗(G,β)∧ is a second countable locally compact Hausdorff space [1, Theorem 1.1.1]. We cite [2, Section 3] to see that each element of Ĝ is of the form (ω,x) with x ∈X and ω a character in the Pontryagin dual of Gx, denoted (Gx). The action of (ω,x) on Cc(G) is given by