Notes on noncommutative ergodic theorems

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