Noncommutative Maximal Ergodic Theorems for Modulated $(C,\alpha)$-Averages with Operator-Valued Weights

Abstract

In this paper, we study modulated $(C,\alpha)$-ergodic averages with operator-valued weights and their subsequential versions in the setting of semifinite von Neumann algebras. Our framework combines $(C,\alpha)$-type modulation with operator-valued weights and extends both scalar-weighted and operator-valued weighted ergodic theories. We establish mean ergodic theorems and maximal inequalities for these averages on noncommutative $L_p$-spaces. We then study bilateral almost uniform and almost uniform convergence in noncommutative Orlicz spaces, including convergence along subsequences of density one. A key ingredient in our approach is the introduction of the $p$-convexity condition on the underlying Orlicz function. This condition enables us to transfer appropriate maximal estimates from noncommutative $L_p$-spaces to establish bilateral uniform equicontinuity in measure at zero on the corresponding noncommutative $p$-convex Orlicz spaces. Together with the noncommutative Banach principle, this yields the desired pointwise convergence results. As an additional consequence of our $p$-convexity argument, we extend Rota's theorem to a larger class of noncommutative Orlicz spaces associated with the $p$-convex Orlicz functions considered in this article.