%0 Journal Article
%T Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
%A Salman Beigi
%A Christopher King
%J Mathematics
%D 2015
%I arXiv
%R 10.1063/1.4934729
%X We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS) inequality for completely bounded norms of one-parameter semigroups of super-operators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross Lemma which allows LS at general $q$ to be deduced from LS at $q=2$.
%U http://arxiv.org/abs/1509.02610v3