Hypercontractivity for free products
Marius Junge, Carlos Palazuelos, Javier Parcet, Mathilde Perrin and, \'Eric Ricard
TL;DR
This paper establishes optimal hypercontractivity bounds for free product extensions of the Ornstein-Uhlenbeck semigroup on Clifford algebras and related structures, generalizing previous results and applying to free Poisson semigroups.
Contribution
It introduces new optimal hypercontractivity bounds for free product extensions of the Ornstein-Uhlenbeck semigroup, extending classical results to Clifford algebras and q-deformed von Neumann algebras.
Findings
Optimal bounds for free product Ornstein-Uhlenbeck semigroup
Hypercontractivity bounds for free Poisson semigroup on free groups
Generalization to q-deformed von Neumann algebras
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 may 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.
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