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    Hypercontractivity on the qq-Araki-Woods algebras

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    Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the qq-Ornstein-Uhlenbeck semigroup on the qq-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 qq-Araki-Woods algebras. We show that hypercontractivity from LpL^p to L2L^2 can occur if and only if the generator of the deformation is bounded.Comment: 17 page
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