Notes on noncommutative ergodic theorems

TL;DR

This paper establishes the completeness of certain noncommutative function spaces under various modes of convergence and extends pointwise Cauchy properties of operator nets to convergence in symmetric spaces, with applications to noncommutative ergodic theory.

Contribution

It proves the completeness of $L^0(\

Findings

Spaces $L^0(\

convergence modes established

Extension of Cauchy property to symmetric spaces

Abstract

Given a semifinite von Neumann algebra $\mathcal M$ equipped with a faithful normal semifinite trace $\tau$, we prove that the spaces $L^0(\mathcal M,\tau)$ and $\mathcal R_\tau$ are complete with respect to pointwise, almost uniform and bilaterally almost uniform, convergences in $L^0(\mathcal M,\tau)$. Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space $L^1(\mathcal M,\tau)$ can be extended to pointwise convergence of such nets in any fully symmetric space $E\subset\mathcal R_\tau$, in particular, in any space $L^p(\mathcal M,\tau)$, $1\leq p<\infty$. Some applications of these results in the noncommutative ergodic theory are discussed.