On the non-commutative Neveu decomposition and stochastic ergodic theorems

TL;DR

This paper proves a non-commutative Neveu decomposition for actions of amenable semigroups and groups on von Neumann algebras, leading to new stochastic ergodic theorems in non-commutative $L^1$-spaces.

Contribution

It establishes the Neveu decomposition for a broad class of actions on semifinite von Neumann algebras, resolving prior open problems and extending ergodic theorems to non-commutative settings.

Findings

Neveu decomposition for amenable semigroup actions on semifinite von Neumann algebras.

Ergodic theorems for positive contractions on non-commutative $L^1$-spaces.

First ergodic theorem beyond the Danford-Schwartz category in non-commutative context.

Abstract

In this article, we prove Neveu decomposition for the action of the locally compact amenable semigroup of positive contractions on semifinite von Neumann algebras and thus, it entirely resolves the problem for the actions of arbitrary amenable semigroup on semifinite von Neumann algebras. We also prove it for amenable group actions by Markov automorphisms on any $\sigma$-finite von Neumann algebras. As an application, we obtain stochastic ergodic theorem for actions of $ \mathbb{Z}_+^d$ and $\mathbb{R}_+^d$ for $ d \in \mathbb{N}$ by positive contractions on $L^1$-spaces associated with a finite von Neumann algebra. It yields the first ergodic theorem for positive contraction on non-commutative $L^1$-spaces beyond the Danford-Schwartz category.