On weak convergence of iterates in quantum \(L_{p}\)-spaces \((p\geq 1)\) (Q2576027)

The paper is devoted to the proof of a theorem on a positive contraction \(T\) in the predual of a von Neumann algebra. Namely, it turns out that weak convergence of the iterates \(\{T^i\}\) is equivalent to strong convergence of the ergodic means \(n^{-1}\sum_{i<n}T^{k_i}\) for each strictly increasing sequence of natural numbers \(\{k_i\}\), and in turn to strong convergence of the sequence \(\{A_n(T)\}\), where \(\{A_n(T)\}=\sum_ia_{n,i}T^i\), for any uniformly regular matrix \((a_{n,i})\). (This theorem has already been published by the second named author with an identical proof in [Vladikavkaz. Mat. Zh. 5, No.~3, 39--45 (electronic) (2003; Zbl 1051.46038)]). The authors also give analogous results for \(L_p\) spaces, \(1<p<\infty\), over a semifinite von Neumann algebra and \(L_p\) spaces, \(1\leq p<\infty\), over a semifinite JBW* algebra.