$T$-admissible processes and noncommutative weighted ergodic theorems

TL;DR

This paper investigates noncommutative ergodic theorems with weighted averages, extending classical methods to noncommutative spaces and including sequences like i.i.d. and M"obius functions.

Contribution

It extends the subsequence argument to noncommutative settings and identifies new classes of weights satisfying decay conditions for ergodic theorems.

Findings

Established b.a.u. convergence for weighted averages with various sequences.

Extended the Wiener-Wintner type ergodic theorem to T-admissible processes.

Included sequences generated by bounded i.i.d. sequences and the M"obius function.

Abstract

In this article, we study the bilaterally almost uniform (b.a.u.) convergence of weighted averages of a positive Dunford-Schwartz operator on the noncommutative $L_p$-spaces associated to a semifinite von Neumann algebra by a large number of weighting sequences. We do this by extending the classical "subsequence argument" to the noncommutative setting. This is then used to establish a large number of sequences satisfying a certain decay condition as good weights for the noncommutative individual ergodic theorem. This class includes those sequences generated by bounded i.i.d. sequences and the M\"{o}bius function. We also study similar problems for $T$-admissible processes on a semifinite von Neumann algebra, showing that if a Wiener-Wintner type ergodic theorem holds for a class $\mathcal{U}\subset W_q$ of weights for $T$-additive process, then it also holds for strongly $p$-bounded $T$-admissible processes, assuming that the duality $\frac{1}{p}+\frac{1}{q}=1$ holds and that $T$ is a normal $\tau$-preserving $*$-automorphism.