Complete Logarithmic Sobolev Inequalities via Ricci Curvature Bounded Below II

TL;DR

This paper establishes complete modified log-Sobolev inequalities for quantum groups using Ricci curvature bounds, demonstrating stability under algebraic operations and applying results to free group factors.

Contribution

It proves the stability of geometric Ricci curvature bounds under tensor and free product operations, extending log-Sobolev inequalities to various quantum group settings.

Findings

Complete modified log-Sobolev inequalities for quantum groups.

Stability of Ricci curvature bounds under tensor and free products.

Application to free group factors and amalgamated free products.

Abstract

Using a non-negative curvature condition, we prove the complete version of modified log-Sobolev inequalities for central Markov semigroups on various compact quantum groups, including group von Neumann algebras, free orthogonal group and quantum automorphism groups. We also prove that the "geometric Ricci curvature lower bound" introduced by Junge-Li-LaRacuente is stable under tensor products and amalgamated free products. As an application, we obtain the geometric Ricci curvature lower bound and complete modified logarithmic Sobolev inequality for word-length semigroups on free group factors and amalgamated free product algebras.