Hypercontractivity for free products

Junge, Marius; Palazuelos, Carlos; Parcet, Javier; Perrin, Mathilde; Ricard, Éric

doi:10.24033/asens.2260

Hypercontractivity for free products
[Hypercontractivité pour des produits libres]

Junge, Marius ; Palazuelos, Carlos ; Parcet, Javier ; Perrin, Mathilde ; Ricard, Éric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 861-889.

Résumé
Abstract

Cet article s'intéresse à des estimations hypercontractives pour des semi-groupes obtenus comme produits libres. Notre approche est basée sur un théorème de la limite centrale pour des produits libres d'algèbres de spin ou autres. Nous obtenons un temps optimal d'hypercontractivité $L_{p} \to L_{q}$ pour les produits libres des semi-groupes d'Orstein-Uhlenbeck sur les algèbres $q$ -déformées ( $- 1 \leq q \leq 1$ ) qui interpolent entre les fermions ( $q = - 1$ ) et les bosons ( $q = 1$ ). Ces résultats s'inspirent des travaux de Nelson, Gross, Carlen/Lieb et Biane et les généralisent. Comme application, nous déduisons un temps d'hypercontractivité $L_{p} \to L_{q}$ pour le semi-groupe de Poisson libre sur l'algèbre du groupe libre à une infinité de générateurs.

In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher's central limit theorem, we can also obtain the same bounds for free products of $q$ -deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields hypercontractivity bounds for the free Poisson semigroup acting on the group algebra of the free group $𝔽_{n}$ , uniformly in the number of generators.

Publié le : 2015-09-24

DOI : 10.24033/asens.2260

Classification : 22D15, 43A22, 47D07, 46L09.
Keywords: Hypercontractivity, Fourier multiplier, group von Neumann algebra, free products.
Mot clés : Hypercontractivité, multiplicateur de Fourier, algèbre de von Neumann, produit libres.

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     author = {Junge, Marius and Palazuelos, Carlos and Parcet, Javier and Perrin, Mathilde and Ricard, \'Eric},
     title = {Hypercontractivity for free products},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {861--889},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {4},
     year = {2015},
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     mrnumber = {3377067},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2260/}
}

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Junge, Marius; Palazuelos, Carlos; Parcet, Javier; Perrin, Mathilde; Ricard, Éric. Hypercontractivity for free products. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 861-889. doi : 10.24033/asens.2260. http://www.numdam.org/articles/10.24033/asens.2260/

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