NOTES ON NONCOMMUTATIVE ERGODIC THEOREMS

Given a semifinite von Neumann algebra M equipped with a faithful normal semifinite trace τ, we prove that the spaces L⁰(M, τ) and Rτ are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L⁰(M, τ). Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L¹(M, τ) can be extended to pointwise convergence of such nets in any fully symmetric space E ⊂ Rτ , in particular, in any space L^p(M, τ), 1 ≤ p < ∞. Some applications of these results in the noncommutative ergodic theory are discussed.