Hypercontractivity on the $q$-Araki-Woods algebras

TL;DR

This paper investigates the hypercontractivity properties of the $q$-Araki-Woods algebras, extending known results from the $q$-deformed free group algebra to the type III setting, with a focus on bounded generators.

Contribution

It establishes conditions under which hypercontractivity from $L^p$ to $L^2$ holds for $q$-Araki-Woods algebras, extending previous work to a new algebraic context.

Findings

Hypercontractivity occurs if and only if the generator is bounded.

Extension of hypercontractivity results from type I to type III algebras.

Characterization of $q$-Araki-Woods algebras in terms of deformation generators.

Abstract

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.