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