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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 4 (2015)

Marius Junge, Carlos Palazuelos, Javier Parcet, Mathilde Perrin, Éric Ricard
Hypercontractivity for free products
Annales scientifiques de l'ENS 48, fascicule 4 (2015), 861-889

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Résumé :
Hypercontractivité pour des produits libres
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 L_q pour les produits libres des semi-groupes d'Orstein-Uhlenbeck sur les algèbres q-déformées (-1q 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 L_q pour le semi-groupe de Poisson libre sur l'algèbre du groupe libre à une infinité de générateurs.

Mots-clefs : Hypercontractivité, multiplicateur de Fourier, algèbre de von Neumann, produit libres.

Abstract:
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.

Keywords: Hypercontractivity, Fourier multiplier, group von Neumann algebra, free products.

Class. math. : 22D15, 43A22, 47D07, 46L09.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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