Noncommutative maximal inequalities for polynomial ergodic averages
Abstract
We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let γ be a trace-preserving automorphism of a semifinite von Neumann algebra (𝒩,τ). We show that the associated polynomial averages ANf:=1/N∑n=1NγP(n)(f), N∈ℕ, satisfy a strong maximal inequality on Lp(𝒩) for every 1<p<∞, extending the previously known restricted range of p. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying L2-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.
Related
- Noncommutative Maximal Ergodic Theorems for Modulated (C,α)-Averages with Operator-Valued Weights
- Pointwise convergence of noncommutative ergodic averages along the primes
- Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces
- Multiple ergodic averages for commuting multiplicative actions
- Pointwise convergence of bilinear ergodic averages for commuting transformations
BibTeX
@misc{hong2026noncommutative,
title = {{Noncommutative maximal inequalities for polynomial ergodic averages}},
author = {Guixiang Hong and Wenbo Li and Eric Ricard and Liang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06455},
eprint = {2610.06455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06455},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncommutative maximal inequalities for polynomial ergodic averages
AU - Guixiang Hong
AU - Wenbo Li
AU - Eric Ricard
AU - Liang Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06455
UR - https://arxiv.org/abs/2610.06455
AB - We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let γ be a trace-preserving automorphism of a semifinite von Neumann algebra (𝒩,τ). We show that the associated polynomial averages A_Nf:=1/N∑_n=1^Nγ^P(n)(f), N∈ℕ, satisfy a strong maximal inequality on L_p(𝒩) for every 1<p<∞, extending the previously known restricted range of p. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying L_2-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.
ER -