Hypercontractivity of heat semigroups on free quantum groups

TL;DR

This paper investigates the hypercontractivity and related functional inequalities of heat semigroups on free quantum groups, providing new insights into their spectral properties and inequalities.

Contribution

It establishes ultracontractivity, hypercontractivity, spectral gap, and logarithmic Sobolev inequalities for semigroups on free quantum groups, advancing understanding of their analytical behavior.

Findings

Semigroups satisfy ultracontractivity and hypercontractivity estimates

Results on spectral gap and logarithmic Sobolev inequalities

Enhanced understanding of quantum group heat semigroup properties

Abstract

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.