Hypercontractivity on the $q$-Araki-Woods algebras

Details Hypercontractivity on the $q$-Araki-Woods algebras Hypercontractivity on the $q$-Araki-Woods algebras

2010-08-31

arxiv.org/abs/1008.5286v1

Mathematics

Operator Algebras

17 pages

Scientific paper

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.

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