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arxiv: 2604.25029 · v1 · submitted 2026-04-27 · 🧮 math.OA · math.DS· math.PR

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Pointwise Convergence for Random Ergodic Averages in Non-commutative L^p-spaces

Christian Le Merdy, Safoura Zadeh

Pith reviewed 2026-05-07 17:06 UTC · model grok-4.3

classification 🧮 math.OA math.DSmath.PR
keywords ergodic averagespointwise convergencenon-commutative Lp spacesvon Neumann algebrasrandom subsequencesbilateral almost uniform convergencepositive contractions
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The pith

Random ergodic averages converge bilaterally almost uniformly in non-commutative Lp spaces

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

The paper proves that for a semifinite von Neumann algebra M and a positive contraction T on its L1 and L∞ spaces, the random averages 1/W_N sum X_n T^n(x) converge almost surely to the ergodic projection. The subsequence is selected by independent Bernoulli random variables with success probability n to the minus alpha, and W_N is the sum of the expected values. Convergence holds in the bilateral almost uniform topology for every element x in Lp(M) when 1 is less than p less than infinity. A sympathetic reader would care because this supplies the non-commutative version of Bourgain's theorem on pointwise convergence along random sparse subsequences.

Core claim

Let M be a semifinite von Neumann algebra and T a positive contraction on both L^1(M) and L^∞(M). We consider ergodic averages along a random sparse subsequence determined by independent Bernoulli variables (X_n) with P(X_n=1)=n^{-α}, and set W_N = sum E[X_n]. We prove that, almost surely, the averages 1/W_N sum X_n T^n(x) converge bilaterally almost uniformly to the ergodic projection for all 1 < p < ∞.

What carries the argument

Bilateral almost uniform convergence, the non-commutative replacement for almost-everywhere pointwise convergence, applied to the randomly thinned iterates of the positive contraction T.

If this is right

  • The stated almost-sure convergence holds for every x in Lp(M) when 1 < p < ∞.
  • The limit is always the ergodic projection associated with T.
  • The result recovers the classical commutative case when M is abelian.

Where Pith is reading between the lines

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

  • The same random-selection method may yield convergence for other classes of subsequences whose densities decay slower or faster than n^{-α}.
  • The estimates could be adapted to study asymptotic behavior of quantum Markov semigroups generated by such contractions.
  • Determining the exact interval of admissible α would constitute a direct quantitative sharpening of the existence result.

Load-bearing premise

The operator T must be positive and contractive simultaneously in the L1 and L∞ norms on the semifinite von Neumann algebra.

What would settle it

A concrete semifinite von Neumann algebra M, positive contraction T, and element x in some Lp(M) for which the random averages fail to converge bilaterally almost uniformly on a set of positive probability in the Bernoulli space.

read the original abstract

Let $M$ be a semifinite von Neumann algebra and $T$ a positive contraction on both $L^1(M)$ and $L^\infty(M)$. We consider ergodic averages along a random sparse subsequence determined by independent Bernoulli variables $(X_n)_{n\geq 1}$ with $\mathbb{P}(X_n = 1) = n^{-\alpha}$, and set $W_N = \sum_{n=1}^N \mathbb{E}[X_n]$. We prove that, almost surely, the averages $\frac{1}{W_N} \sum_{n=1}^N X_n\, T^n(x)$ converge bilaterally almost uniformly to the ergodic projection for all $1 < p < \infty$. This extends a theorem of Bourgain to the non-commutative setting.

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

1 major / 2 minor

Summary. The manuscript proves that for a semifinite von Neumann algebra M and a positive contraction T on both L^1(M) and L^∞(M), the random ergodic averages (1/W_N) ∑_{n=1}^N X_n T^n(x) converge bilaterally almost uniformly to the ergodic projection almost surely, for all 1 < p < ∞. Here the X_n are independent Bernoulli random variables with P(X_n = 1) = n^{-α} and W_N = ∑ E[X_n]. This is presented as a direct extension of Bourgain's random ergodic theorem to the non-commutative setting.

Significance. If correct, the result is a solid extension of a classical theorem into non-commutative L^p spaces. It employs the standard Dunford-Schwartz assumptions on T and the appropriate notion of bilateral almost uniform convergence, which is the natural topology for pointwise results in this setting. The work could serve as a foundation for further developments in non-commutative ergodic theory and random dynamical systems on von Neumann algebras.

major comments (1)
  1. [Abstract / Theorem statement] Abstract and main theorem statement: the range of the parameter α is left unspecified. This is load-bearing for the central claim, as the almost-sure convergence and the applicability of the requisite non-commutative maximal inequalities and law-of-large-numbers estimates for the random subsequence depend on α lying in a specific interval (typically 0 < α < 1); without an explicit range the scope of the theorem remains unclear.
minor comments (2)
  1. [Introduction] The notation for bilateral almost uniform convergence and the ergodic projection should be recalled briefly in the introduction or preliminaries for accessibility.
  2. [References] Ensure the citation to Bourgain's original theorem includes the precise reference and year.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The single major comment is addressed below; we agree that explicit specification of the parameter range improves clarity and will revise accordingly.

read point-by-point responses

  1. Referee: Abstract and main theorem statement: the range of the parameter α is left unspecified. This is load-bearing for the central claim, as the almost-sure convergence and the applicability of the requisite non-commutative maximal inequalities and law-of-large-numbers estimates for the random subsequence depend on α lying in a specific interval (typically 0 < α < 1); without an explicit range the scope of the theorem remains unclear.

    Authors: We agree that the range must be stated explicitly. The proof relies on 0 < α < 1 to guarantee that W_N → ∞ almost surely (by the law of large numbers for independent Bernoulli variables) while ensuring sufficient sparsity for the non-commutative maximal inequalities to apply. We will add the explicit hypothesis 0 < α < 1 to the abstract, the introduction, and the statement of the main theorem in the revised manuscript. This is a clarification only and does not change the argument or the result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external result independently

full rationale

The paper states a direct extension of Bourgain's random ergodic theorem to the non-commutative L^p setting for semifinite von Neumann algebras, using the standard Dunford-Schwartz assumptions on the positive contraction T and proving bilateral almost uniform convergence. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the claim or structure. The Bernoulli subsequence and W_N normalization are defined externally from the input probabilities, and the ergodic projection is the standard fixed-point operator. The result is self-contained against external benchmarks in non-commutative ergodic theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions from non-commutative analysis and the specific random Bernoulli model. No free parameters are fitted to data, no new entities are postulated, and no ad-hoc axioms beyond the semifinite von Neumann algebra framework are visible in the abstract.

axioms (2)
  • domain assumption M is a semifinite von Neumann algebra
    Required to define the non-commutative L^p spaces in which the averages live.
  • domain assumption T is a positive contraction on L^1(M) and L^∞(M)
    Ensures the iterates T^n are well-defined and contractive so that the averages make sense.

pith-pipeline@v0.9.0 · 5442 in / 1386 out tokens · 71452 ms · 2026-05-07T17:06:22.306182+00:00 · methodology

discussion (0)

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Reference graph

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