Notes on noncommutative ergodic theorems

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 $L^p(\mathcal{M},\tau )$, $1\le p<\mathrm{\infty }$. Some applications of these results in the noncommutative ergodic theory are discussed.