Journal of Operator Theory

Volume 77, Issue 1, Winter 2017 pp. 61-76.

Hypercontractivity of heat semigroups on free quantum groups

Authors: Uwe Franz (1), Guixiang Hong (2), Francois Lemeux (3), Michael Ulrich (4), and Haonan Zhang (5)
Author institution:(1) Laboratoire de Mathematiques de Besancon, Universite de Bourgogne-Franche-Comte, France
(2) School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
(3) Laboratoire de Mathematiques de Besancon, Universite de Bourgogne-Franche-Comte, France
(4) Laboratoire de Mathematiques de Besancon, Universite de Bourgogne-Franche-Comte, France
(5) Laboratoire de Mathematiques de Besancon, Universite de Bourgogne-Franche-Comte, France

Summary: In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.

DOI: http://dx.doi.org/10.7900/jot.2015nov13.2126
Keywords: free quantum group, heat semigroup, hypercontractivity, logarithmic Sobolev inequality

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