Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups

Panchugopal Bikram, Hariharan G, Sudipta Kundu, Diptesh Saha

TL;DR

The article develops non-commutative maximal inequalities and pointwise ergodic theorems for state-preserving actions of amenable groups on tracial $L^1$-spaces associated with von Neumann algebras. It leverages a tracial-to-state transfer via a non-commutative maximal inequality and a Banach principle to establish bilaterally almost uniform convergence of ergodic averages for functionals in $\mathcal{M}_*$, and extends to stochastic ergodic behavior using the Neveu decomposition. The sharp maximal bounds avoid extra error terms present in earlier work, and the results cover unimodular amenable groups with admissible Følner sequences, yielding both convergence in measure and a.u. limits. Collectively, these contributions advance non-commutative ergodic theory for group actions, enabling stochastic descriptions of long-term behavior in operator-algebraic dynamical systems.

Abstract

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.

Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups

TL;DR

The article develops non-commutative maximal inequalities and pointwise ergodic theorems for state-preserving actions of amenable groups on tracial -spaces associated with von Neumann algebras. It leverages a tracial-to-state transfer via a non-commutative maximal inequality and a Banach principle to establish bilaterally almost uniform convergence of ergodic averages for functionals in , and extends to stochastic ergodic behavior using the Neveu decomposition. The sharp maximal bounds avoid extra error terms present in earlier work, and the results cover unimodular amenable groups with admissible Følner sequences, yielding both convergence in measure and a.u. limits. Collectively, these contributions advance non-commutative ergodic theory for group actions, enabling stochastic descriptions of long-term behavior in operator-algebraic dynamical systems.

Abstract

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative -spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.
Paper Structure (7 sections, 24 theorems, 133 equations)

This paper contains 7 sections, 24 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-commutative $L^1$-spaces
  4. Non-commutative dynamical systems
  5. Maximal Inequality
  6. Pointwise ergodic theorems
  7. Stochastic Ergodic Theorem

Key Result

Theorem 1.1

Let $\phi \in \mathcal{M}_*$, then consider the averages $(A_k(\phi))_{k \in \mathbb{N}}$, where Then the following hold:

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 39 more