A mean ergodic theorem in von-Neumann algebras

TL;DR

This paper extends the mean ergodic theorem to von-Neumann algebras and their pre-dual spaces, establishing new dualities and proving ergodic theorems for group actions on these spaces.

Contribution

It introduces a duality between von-Neumann's and Birkhoff's mean ergodic theorems and proves Birkhoff's theorem for amenable group actions on pre-dual Banach spaces.

Findings

Improved mean ergodic theorems for von-Neumann algebras

Established Birkhoff's mean ergodic theorem for amenable group actions

Demonstrated duality between ergodic theorems in von-Neumann algebras and their pre-duals

Abstract

We explore a duality between von-Neumann's mean ergodic theorem in von-Neumann algebra and Birkhoff's mean ergodic theorem in the pre-dual Banach space of von-Neumann algebras. Besides improving known mean ergodic theorems on von-Neumann algebras, we prove Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space.