$T$-admissible processes and noncommutative weighted ergodic theorems

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. A version of this result for arbitrary positive Dunford-Schwartz operators is also considered.