Logarithmic Sobolev inequalities and exponential entropy decay in non-commutative algebras

TL;DR

This paper explores the connections between logarithmic Sobolev inequalities, entropy decay, and spectral gaps in non-commutative algebras, establishing implications and equivalences under certain conditions.

Contribution

It proves that non-commutative log-Sobolev inequalities imply spectral gap inequalities and characterizes exponential entropy decay as equivalent to a modified log-Sobolev inequality.

Findings

Log-Sobolev inequalities imply spectral gap inequalities with optimal constants.

Exponential entropy decay is equivalent to a modified log-Sobolev inequality.

Regularity conditions on quadratic forms ensure entropy decay from log-Sobolev inequalities.

Abstract

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities; this entropy decay turns out to be implied by the usual log-Sobolev inequality adding some regularity conditions on the quadratic forms.