Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm

TL;DR

This paper extends hypercontractivity and log-Sobolev inequalities to completely bounded norms of semigroups on matrix algebras, establishing their equivalence and a generalized Gross Lemma.

Contribution

It introduces the notions of HC and LS for completely bounded norms and proves their equivalence, along with a generalized Gross Lemma for these inequalities.

Findings

Established the equivalence of HC and LS for completely bounded norms.

Proved a version of the Gross Lemma for general q.

Extended classical inequalities to the setting of matrix algebra semigroups.

Abstract

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 $q=2$.