Notes on noncommutative ergodic theorems

Abstract

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