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arxiv: 2604.26224 · v1 · submitted 2026-04-29 · 🧮 math.OA · math.DS

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T-admissible processes and noncommutative weighted ergodic theorems

Morgan O'Brien

Pith reviewed 2026-05-07 12:48 UTC · model grok-4.3

classification 🧮 math.OA math.DS
keywords noncommutative ergodic theoremsweighted averagessubsequence argumentT-admissible processesvon Neumann algebrasMöbius functionbilaterally almost uniform convergenceDunford-Schwartz operators
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The pith

The classical subsequence argument extends to noncommutative L_p spaces, making sequences from bounded i.i.d. processes and the Möbius function good weights for bilaterally almost uniform convergence of weighted ergodic averages.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that extending the subsequence argument to the noncommutative setting identifies many sequences obeying a decay condition as valid weights for the individual ergodic theorem on noncommutative L_p-spaces of semifinite von Neumann algebras. This class explicitly includes weights generated by bounded i.i.d. sequences and by the Möbius function. The work also shows that Wiener-Wintner-type results for a subclass of weights transfer to strongly p-bounded T-admissible processes whenever duality 1/p + 1/q = 1 holds and T is a normal tau-preserving *-automorphism. A sympathetic reader cares because these extensions enlarge the set of applicable weighting sequences beyond what was previously known in the noncommutative case, directly supporting convergence statements for positive Dunford-Schwartz operators.

Core claim

By adapting the classical subsequence argument to bilaterally almost uniform convergence, the paper proves that any sequence satisfying the required decay condition serves as a good weight for the noncommutative individual ergodic theorem. Concrete members of this class are those arising from bounded i.i.d. sequences and from the Möbius function. For T-admissible processes the same decay condition plus a Wiener-Wintner assumption on a dense subclass of weights in W_q implies the corresponding theorem for strongly p-bounded T-admissible processes, provided 1/p + 1/q = 1 and T is normal and tau-preserving.

What carries the argument

the extension of the classical subsequence argument to bilaterally almost uniform convergence for positive Dunford-Schwartz operators acting on noncommutative L_p-spaces

If this is right

  • Bounded i.i.d. sequences generate good weights for the noncommutative individual ergodic theorem.
  • The Möbius function generates good weights for the same theorem.
  • Wiener-Wintner results for a suitable subclass of weights transfer to all strongly p-bounded T-admissible processes under the stated duality and normality conditions.
  • The same decay condition suffices for a large family of weighting sequences in the noncommutative setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These weights may now be used to obtain pointwise convergence results in quantum dynamical systems modeled by semifinite von Neumann algebras.
  • The transfer mechanism for T-admissible processes suggests analogous statements could hold for other classes of weights once a Wiener-Wintner theorem is verified for a dense subclass.
  • Arithmetic sequences such as the Möbius function becoming admissible opens the possibility of linking noncommutative ergodic theory to questions in analytic number theory.

Load-bearing premise

The classical subsequence argument extends rigorously to bilaterally almost uniform convergence in the noncommutative L_p setting for semifinite von Neumann algebras.

What would settle it

A concrete sequence obeying the decay condition together with a positive Dunford-Schwartz operator on some noncommutative L_p-space for which the weighted averages fail to converge bilaterally almost uniformly.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the classical subsequence argument to prove bilaterally almost uniform (b.a.u.) convergence of weighted averages for positive Dunford-Schwartz operators on noncommutative L_p-spaces of semifinite von Neumann algebras. It identifies a broad class of sequences satisfying a decay condition—including those generated by bounded i.i.d. sequences and the Möbius function—as good weights for the noncommutative individual ergodic theorem. It further establishes that Wiener-Wintner-type results for T-additive processes transfer to strongly p-bounded T-admissible processes when 1/p + 1/q = 1 and T is a normal τ-preserving *-automorphism.

Significance. If the noncommutative extension of the subsequence argument holds with the required control on projections, the work would meaningfully broaden weighted ergodic theorems beyond the commutative setting, providing new tools for noncommutative probability and operator algebras. The explicit inclusion of arithmetic sequences such as the Möbius function strengthens the applicability, and the transfer result for T-admissible processes offers a useful duality-based reduction.

major comments (2)
  1. [Main theorem on b.a.u. convergence (around the extension of the subsequence argument)] The central extension of the subsequence argument to b.a.u. convergence (detailed in the proof of the main weighted ergodic theorem) must explicitly construct the controlling projections e_n such that τ(e_n) → 0 while ensuring ||(A_n^w - A) x e_n||_p → 0 for the weighted averages. The classical Borel-Cantelli approach relies on pointwise maximal functions, which are unavailable here; without additional estimates on how the weights interact with the semifinite trace and the positive Dunford-Schwartz operator, the uniform control on the remainder may not follow directly.
  2. [Section on T-admissible processes and Wiener-Wintner transfer] In the transfer result for T-admissible processes, the assumption that a Wiener-Wintner theorem for U ⊂ W_q implies the result for strongly p-bounded T-admissible processes (via duality 1/p + 1/q = 1 and normality of T) requires verification that strong p-boundedness preserves the necessary maximal inequalities when passing to the dual weights. The current argument appears to inherit the same projection-control issue from the first part without additional noncommutative estimates.
minor comments (2)
  1. [Introduction / main theorem statement] The decay condition on the weighting sequences is stated in the abstract but should be recalled explicitly with its precise formulation (e.g., the summability or decay rate) at the beginning of the main theorem statement for readability.
  2. [Preliminaries and definitions] Notation for bilaterally almost uniform convergence and the projections e_n should be standardized across sections to avoid minor inconsistencies in the bilateral versus unilateral versions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on noncommutative weighted ergodic theorems. We address each major comment below with clarifications on our proof techniques.

read point-by-point responses

  1. Referee: [Main theorem on b.a.u. convergence (around the extension of the subsequence argument)] The central extension of the subsequence argument to b.a.u. convergence (detailed in the proof of the main weighted ergodic theorem) must explicitly construct the controlling projections e_n such that τ(e_n) → 0 while ensuring ||(A_n^w - A) x e_n||_p → 0 for the weighted averages. The classical Borel-Cantelli approach relies on pointwise maximal functions, which are unavailable here; without additional estimates on how the weights interact with the semifinite trace and the positive Dunford-Schwartz operator, the uniform control on the remainder may not follow directly.

    Authors: We thank the referee for this observation on the key technical step. In the proof of our main weighted ergodic theorem, the subsequence argument is extended by first applying the noncommutative maximal inequality for positive Dunford-Schwartz operators (which is available in the semifinite setting) to select a subsequence along which the weighted averages converge in norm on a reduced algebra. The controlling projections e_n are constructed explicitly as the spectral projections of the maximal function associated to the weighted averages, with τ(1 - e_n) decaying to zero by the trace property and the given decay condition on the weights (which controls the increments ||A_{n+1}^w - A_n^w|| via summation). The norm convergence ||(A_n^w - A) x e_n||_p → 0 then follows by bounding the remainder on e_n L_p e_n using the operator's positivity and the semifinite trace to replace the classical Borel-Cantelli summation. We agree that the current exposition would benefit from a more detailed, step-by-step construction of these e_n and the interaction with the weights; we will revise the proof to include this explicit construction and the necessary estimates. revision: partial

  2. Referee: [Section on T-admissible processes and Wiener-Wintner transfer] In the transfer result for T-admissible processes, the assumption that a Wiener-Wintner theorem for U ⊂ W_q implies the result for strongly p-bounded T-admissible processes (via duality 1/p + 1/q = 1 and normality of T) requires verification that strong p-boundedness preserves the necessary maximal inequalities when passing to the dual weights. The current argument appears to inherit the same projection-control issue from the first part without additional noncommutative estimates.

    Authors: We appreciate the referee's point on the transfer theorem for T-admissible processes. The argument proceeds by duality: given a Wiener-Wintner result for weights in U ⊂ W_q, the strong p-boundedness of the T-admissible process (with 1/p + 1/q = 1) ensures that the associated maximal operator remains bounded when transferred to the dual space via the normal τ-preserving *-automorphism T. This boundedness, combined with the same subsequence argument as in the first part, yields the projection control for the dual weights. However, we acknowledge that the write-up would be strengthened by an additional lemma or remark explicitly verifying that strong p-boundedness preserves the maximal inequalities under this duality. We will incorporate such a clarification in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; extends external classical subsequence argument

full rationale

The derivation extends the classical subsequence argument (an external technique) to bilaterally almost uniform convergence for positive Dunford-Schwartz operators on noncommutative L_p spaces, then applies it to establish good weights under a decay condition. The T-admissible process result is conditional on an assumed Wiener-Wintner theorem for a subclass of weights plus standard duality and normality assumptions. No steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations; the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the work rests on standard domain assumptions of noncommutative analysis rather than new fitted parameters or invented entities.

axioms (2)
  • domain assumption Semifinite von Neumann algebras admit noncommutative L_p spaces on which positive Dunford-Schwartz operators act as contractions suitable for ergodic averages.
    Invoked when defining the setting for weighted averages and b.a.u. convergence.
  • domain assumption The duality relation 1/p + 1/q = 1 holds and T is a normal tau-preserving *-automorphism when transferring results to T-admissible processes.
    Stated explicitly as a condition for the Wiener-Wintner transfer.

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Reference graph

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