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Hypercontractivity for free products

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arxiv 1211.4759 v2 pith:BKDQULX2 submitted 2012-11-20 math.OA math.FAmath.PR

Hypercontractivity for free products

classification math.OA math.FAmath.PR
keywords freeboundshypercontractivityproductsactingalgebraalgebrascentral
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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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