Debabrata De | Searcharxiv

arXiv2026

A characterization of purely atomic von Neumann algebras

Let $M$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $τ$, and let $Φ$ be an Orlicz function. We prove that the noncommutative Orlicz space $L^Φ(M,τ)$ is contained in $M$ if and only if $M$ is purely atomic and there is $δ>0$ such that $τ(e)\geqδ$ for every nonzero projection $e\in M$. We also prove that every left $M$-module map on $L^Φ(M,τ)$ is bounded. As a consequence, a left ideal of $M$ contained in $L^Φ(M,τ)$ and dense in it admits an $M$-invariant algebraic complement if and only if it is equal to $L^Φ(M,τ)$. These results extend a theorem of Fidaleo and Zsidó from $L^2$ to all noncommutative Orlicz spaces.

math.OA↗

arXiv2026

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

In this paper, we study modulated $(C,α)$-ergodic averages with operator-valued weights and their subsequential versions in the setting of semifinite von Neumann algebras. Our framework combines $(C,α)$-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.

math.OA↗