Une généralisation de la notion de transformée de Fourier-Stieltjes

Herz, Carl S.

doi:10.5802/aif.522

Herz, Carl S.

Annales de l'Institut Fourier, Tome 24 (1974) no. 3, pp. 145-157.

Résumé
Abstract

L’espace $P F_{p} (G)$ des $p$ -pseudofonctions sur un groupe localement compact $G$ est le complété de $L_{1} (G)$ pour la norme de convoluteur de $L_{p} (G)$ . Dans le cas où le groupe $G$ est moyennable alors le banach dual à $P F_{p} (G)$ s’identifie avec une certaine algèbre $B_{p} (G)$ de fonctions continues sur $G$ . L’algèbre $B_{p} (G)$ est déjà connue mais ici on montre que $B_{p}$ est un foncteur de groupes localement compacts. Pour $p = 2$ alors $P F_{2} (G)$ est l’algèbre $C^{*}$ de $G$ dont le dual est $F S (G)$ , l’algèbre de transformées de Fourier-Stieltjes. Donc, pour un groupe moyennable, élément de $B_{n} (G)$ généralise la notion de transformée de Fourier-Stieltjes avec coïncidence des deux notions en cas $p = 2$ .

The space $P F_{p} (G)$ of $p$ -pseudofunctions on a locally compact group $G$ is the completion of $L_{1} (G)$ for the norm of convolvers of $L_{p} (G)$ . In case the group $G$ is amenable, the dual Banach space of $P F_{p} (G)$ may be identified with a certain algebra $B_{p} (G)$ of continuous functions on $G$ . The algebra $B_{p} (G)$ is already known, but here it is shown that $B_{p}$ is a functor of locally compact groups. When $p = 2$ we have that $P F_{2} (G)$ is the $C^{*}$ -algebra of $G$ whose dual is $F S (G)$ , the algebra of Fourier-Stieltjes transforms. Thus for an amenable group, element of $B_{n} (G)$ generalizes the notion of Fourier-Stieltjes transform with which it coincides in case $p = 2$ .

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.522

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     title = {Une g\'en\'eralisation de la notion de transform\'ee de {Fourier-Stieltjes}},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {3},
     year = {1974},
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Herz, Carl S. Une généralisation de la notion de transformée de Fourier-Stieltjes. Annales de l'Institut Fourier, Tome 24 (1974) no. 3, pp. 145-157. doi : 10.5802/aif.522. https://www.numdam.org/articles/10.5802/aif.522/

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