$T$-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 Möbius 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 $τ$-preserving $*$-automorphism.

• The paper introduces a comprehensive framework that extends good weight criteria to noncommutative Lp spaces and ensures b.a.u. convergence.
• It adapts the classical subsequence argument to establish weighted convergence under exponential decay conditions for a wide variety of weight sequences.
• It generalizes T-additive to T-admissible processes, accommodating bounded error terms and expanding applications in operator-algebraic dynamics.

Summary of "$T$-admissible processes and noncommutative weighted ergodic theorems" (2604.26224)

Introduction and Context

This article develops the theory of weighted ergodic theorems in the framework of noncommutative $L_p$-spaces associated with semifinite von Neumann algebras, focusing on a broad class of weighting sequences and their interaction with positive Dunford-Schwartz operators. It extends classical results regarding good weights for individual ergodic theorems to the noncommutative regime and significantly enlarges the set of admissible weighting sequences—previously very restrictive in the noncommutative case. The work also generalizes ergodic convergence from $T$-additive processes to the more versatile class of $T$-admissible processes, providing new results on their weighted ergodic convergence.

Noncommutative Weighted Ergodic Theorems

General Convergence Method

A central technical contribution is the adaptation of the classical "subsequence argument" to noncommutative $L_p$-spaces. The paper rigorously establishes that b.a.u. (bilaterally almost uniform) convergence of weighted averages is implied for a large class of weight sequences, as long as exponential averages of the weights obey a summability condition along appropriate lacunary subsequences. Formally, for a bounded sequence $\alpha=(\alpha_k)$,

$$\sum_{m=1}^{\infty}\sup_{\lambda\in\mathbb{T}}|\widehat{M}_{\rho^m}^\alpha(\lambda)|^2<\infty$$

for some lacunary sequence $(\rho^m)$ ensures $\alpha$ is a good weight for the individual ergodic theorem on noncommutative $L_p$ spaces. This condition encompasses classical weights (unit circle powers, bounded Besicovitch, Hartman almost periodic, Möbius, Liouville, automatic, $L_p$0-multiplicative, and weights generated by bounded i.i.d. sequences) and now extends to the noncommutative setting.

Applications to Weighted Sequences

By translating classical maximal inequalities and moment conditions to the operator setting, the paper demonstrates that:

• Bounded i.i.d. sequences: For almost every realization, weighted ergodic averages converge b.a.u. in $L_p$1 for any positive Dunford-Schwartz operator, with the limit proportional to the expectation.
• i.i.d. sequences with finite $L_p$2-th moment ($L_p$3): Under duality constraints ($L_p$4), the same convergence is valid even for unbounded weights.
• Arithmetic weights (e.g., Möbius, Liouville): Convergence follows directly from classical bounds (e.g., Davenport's estimate).
• Automatic and $L_p$5-multiplicative sequences: The framework brings several cases previously only known in the commutative theory into the noncommutative field.

A crucial result is that the b.a.u. convergence always yields limit zero in the $L_p$6-norm for weights satisfying the above exponential decay, establishing a strong parallel to classical return time phenomena.

$L_p$7-admissible Processes and Their Weighted Ergodic Properties

Extension from $L_p$8-additive to $L_p$9-admissible

The paper introduces the concept of strongly $T$0-bounded $T$1-admissible processes $T$2, generalizing additive processes. The core finding is that if a Wiener-Wintner type theorem holds for a class $T$3 of weights and $T$4 is a normal $T$5-preserving $T$6-automorphism, then the same weighted ergodic convergence holds for every strongly $T$7-bounded $T$8-admissible process, under standard Hölder duality.

This is established via dominant process techniques and a noncommutative Banach principle, showing convergence for averages $T$9 for all $T$0. The result allows for the systematic application of weighted ergodic theorems to wider classes than previously possible, including processes governed by recurrence phenomena, growing sets (e.g., return times in dynamical systems), and processes with additive or bounded error.

Further Generalizations

The article also shows that if a $T$1-admissible sequence is subject to additional controlled error terms (having summable $T$2-norm), it can be decomposed into a difference of two strongly $T$3-bounded $T$4-admissible processes. This reduction preserves the weighted ergodic convergence, further expanding the applicability in noncommutative ergodic theory.

Implications and Directions

The extension of the classical subsequence argument to the noncommutative setting opens the way for new investigation into convergence phenomena under irregular, arithmetic, or stochastic weights for operator-algebraic dynamical systems. This work removes artificial barriers separating commutative and noncommutative ergodic convergence and provides comprehensive maximal inequalities and convergence criteria that operate under general conditions, reflecting wider applicability in mathematical physics, noncommutative dynamical systems, and quantum probability.

The results on $T$5-admissible processes suggest that ergodic properties in noncommutative $T$6 spaces are substantially more robust than previously expected, making weighted convergence available for processes that arise in general operator setups—not only strict automorphisms but also processes with additive or subadditive structure.

Future research can use these criteria for weighted convergence to study spectral properties, noncommutative evolution equations, return time phenomena, and even aspects of algorithmic randomness in quantum systems.

Conclusion

The paper substantially enlarges the catalogue of good weights for noncommutative ergodic theorems, establishing rigorous weighted convergence results for a wide variety of sequences. Importantly, it demonstrates that weighted ergodic convergence for positive Dunford-Schwartz operators and normal $T$7-preserving $T$8-automorphisms extends to strongly $T$9-bounded $L_p$0-admissible processes, including those with controlled error terms. The theoretical advancements provided here remove a number of commutative/noncommutative dichotomies and set the stage for further investigations in operator-algebraic ergodic theory and its applications.