The Global Topology of Pontrjagin Duality

TL;DR

This paper extends Pontrjagin duality within fibre bundle frameworks, defining a Fourier transform via duality triples that generalizes classical group $C^*$-algebra isomorphisms.

Contribution

It introduces a novel fibre bundle-based approach to Pontrjagin duality, establishing a generalized Fourier transform and isomorphism of Hilbert $C^*$-modules.

Findings

Defines a Fourier transform using Pontrjagin duality triples

Establishes an isomorphism of Hilbert $C^*$-modules

Generalizes classical duality between group $C^*$-algebras and functions

Abstract

Pontrjagin duality is implemented in the framework of fibre bundles. By means of Pontrjagin duality triples a Fourier transform is defined by a pull-push construction operating on sections of line bundles. This yields an isomorphism of Hilbert $C^*$-modules which generalises the classical isomorphism between the group $C^*$-algebra of a group and the continuous functions vanishing at infinity on the dual group.