Introduction

Michel Enock and Jean-Marie Schwartz
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Michel Enock: Université Pierre et Marie Curie, CNRS, Laboratoire de Mathématiques Fondamentales
Jean-Marie Schwartz: Université Pierre et Marie Curie, CNRS, Laboratoire de Mathématiques Fondamentales

A chapter in Kac Algebras and Duality of Locally Compact Groups, 1992, pp 1-6 from Springer

Abstract: Abstract In the early nineteen thirties, L.S. Pontrjagin ([121]) established his famous duality theorem for abelian locally compact groups: he showed that the set of characters of an abelian locally compact group G is an abelian group, called the dual group of G, and noted Ĝ, which is locally compact for a suitable topology. Moreover, the bidual Gˆˆ is canonically isomorphic and homeomorphic to G; the Fourier transform carries the convolution algebra L 1 (G) into the multiplication algebra L∞(Ĝ); conversely, the Fourier inverse transform carries the convolution algebra L 1 (Ĝ) into the multiplication algebra L∞(G). These two transformations are transposed of each other in the following scheme: $${L^1}(G)\mathop \to \limits^{Fourier} {L^\infty }(\mathop G\limits^ \wedge) $$ $${L^\infty }(G)\mathop \leftarrow \limits^{Fourierinverse} {L^1}(\mathop G\limits^ \wedge) $$ Since this fundamental result, which can be considered as one of the founding theorems of abstract harmonic analysis, a series of duality theorems for nonabelian locally compact groups has been gradually set up, but these dualities were not formulated within the category of locally compact groups.

Keywords: Irreducible Representation; Hopf Algebra; Quantum Group; Compact Group; Duality Theorem (search for similar items in EconPapers)
Date: 1992
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DOI: 10.1007/978-3-662-02813-1_1

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