Hypercontractivity for free products

Title: Hypercontractivity for free products.

Author(s): Marius Junge, Carlos Palazuelos, Javier Parcet, Mathilde Perrin, Éric Ricard.

Source: Annales scientifiques de l’ENS 48, fascicule 4 (2015), 861-889

Date of publication: 2015.

Abstract: In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbecksemigroup 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 Fn, uniformly in the number of generators.

Link: http://smf4.emath.fr/Publications/AnnalesENS/4_48/html/ens_ann-sc_48_861-889.php