Group Bundle Duality

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 {β} 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 {β}. Then C(G, β) is a separable abelian Calgebra 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 (1) (ω, x)(f) = ∫