Introduction

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.