Lamperti Operators, Dilation Theory, and Applications in Noncommutative Ergodic Theory

TL;DR

This paper develops a new framework for quantitative mean ergodic theorems in noncommutative spaces, utilizing advanced harmonic analysis and dilation techniques to extend classical results to noncommutative $L_p$-spaces and semigroup actions.

Contribution

It introduces a novel approach combining non-homogeneous harmonic analysis and dilation theory to establish quantitative ergodic theorems for noncommutative $L_p$-spaces and commuting operator tuples.

Findings

Proved square function inequalities for noncommutative ergodic averages.

Established endpoint estimates for noncommutative square functions.

Extended dilation theorems to commuting tuples on Banach spaces, including noncommutative $L_p$-spaces.

Abstract

In this paper, we develop a novel framework for quantitative mean ergodic theorems in the noncommutative setting, with a focus on actions of amenable groups and semigroups. We prove square function inequalities for ergodic averages arising from actions of groups of polynomial volume growth on a fixed noncommutative $L_p$-space for $1<p<\8$. To achieve this, we establish two endpoint estimates for a noncommutative square function on non-homogeneous space. Our approach relies on semi-commutative non-homogeneous harmonic analysis, including the non-doubling Calder\'on-Zygmund arguments for non-smooth kernels and $\mathrm{BMO}$ space theory, operator-valued inequalities related to balls and cubes in groups equipped with non-doubling measures, and a noncommutative generalization of the classical transference method for amenable group actions. As an application, we establish a quantitative ergodic theorem for the ergodic averages associated with the positive power of modulus representation arising from a Lamperti representation on noncommutative $L_p$-spaces, extending some results in \cite{Templeman2015}. To obtain quantitative ergodic theorem for semigroups of operators, in this paper, we address the open question of extending dilation theorem of Fackler-Gl\"uck from single operators to commuting tuples on Banach spaces including noncommutative $L_p$-spaces. Indeed our approach provides genuine joint $N$-dilations for commuting families, unifying and extending the classical dilation theorems of Sz.-Nagy--Foia\c{s} and Ak\c{c}oglu--Sucheston for a natural class of commuting tuple of contractions extending the abstract dilation theorem of of Fackler-Gl\"uck for commuting tuple of contractions. This enables us to obtain a quantitative ergodic theorem for a large class of semigroups of operators on $\mathbb{R}^d_{+}$.