$T$-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 $T$ acting on the noncommutative $L_p$-spaces $L_p(\mathcal{M},\tau )$ associated to a semifinite von Neumann algebra $(\mathcal{M},\tau )$ for every $1\le p<\infty$ for a variety of sequences which includes those generated by bounded i.i.d. sequences, the Möbius function, the Liouville function, and totally balanced automatic sequences. If $1<p<\infty$, then sequences generated by i.i.d. sequences with finite $q$-th moment for $q=\frac{p}{p-1}$ being the Hölder conjugate of $p$ are also shown to generate good weights for $L_p(\mathcal{M},\tau )$. 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 $\mathcal{W}\subset W_q$ of weights, 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 $\ast$-automorphism. A version of this result for arbitrary positive Dunford-Schwartz operators is also considered.

arXiv comment30 pages. v3: Minor stylistic changes and rephrasings. Added Theorem 3.4 to emphasize a few applications of the main method already mentioned earlier in the article