Local ergodic theorems in symmetric spaces of measurable operators

TL;DR

This paper proves local ergodic theorems for actions of the positive real semigroup in symmetric spaces of measurable operators linked to semifinite von Neumann algebras, advancing the understanding of convergence behaviors in non-commutative analysis.

Contribution

It introduces local mean and individual ergodic theorems for semigroup actions in symmetric operator spaces, extending ergodic theory to non-commutative settings.

Findings

Established local mean ergodic theorems.

Proved individual ergodic theorems with almost uniform convergence.

Extended ergodic results to symmetric spaces of measurable operators.

Abstract

Local mean and individual (with respect to almost uniform convergence in Egorov's sense) ergodic theorems are established for actions of the semigroup $\mathbb R_+^d$ in symmetric spaces of measurable operators associated with a semifinite von Neumann algebra.