Pointwise convergence of noncommutative ergodic averages along the primes

Abstract

Let 𝒩 be a von Neumann algebra equipped with a normal faithful semifinite trace, and let γ be a trace-preserving automorphism of 𝒩. We consider the ergodic averages along the prime numbers A_N(x) := 1/|P_N| ∑_{q∈ P_N}γ^q(x), P_N:={q≤ N:q is prime}. For every 1<p<∞, we prove a strong maximal inequality for (A_N)_N≥2 on L_p(𝒩) and that A_N(x) converges bilaterally almost uniformly for every x∈ L_p(𝒩). The proof exploits the circle method and a noncommutative sampling principle. For the convergence result, Bourgain's commutative argument uses pointwise maximal functions and exceptional sets. These tools are not available in the noncommutative setting. Instead, we show that the tails of the ergodic averages tend to zero in L₂(𝒩;ℓ_∞) and that the difference from the limit belongs to L₂(𝒩;c₀). This gives the desired b.a.u. convergence, and provides a positive answer to one question left open in \cite{ChenHongWang+arXiv2024}.

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BibTeX RIS

BibTeX

@misc{hong2026pointwise,
  title = {{Pointwise convergence of noncommutative ergodic averages along the primes}},
  author = {Guixiang Hong and Liang Wang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30339},
  eprint = {2608.30339},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30339},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Pointwise convergence of noncommutative ergodic averages along the primes
AU  - Guixiang Hong
AU  - Liang Wang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30339
UR  - https://arxiv.org/abs/2608.30339
AB  - Let 𝒩 be a von Neumann algebra equipped with a normal faithful semifinite trace, and let γ be a trace-preserving automorphism of 𝒩. We consider the ergodic averages along the prime numbers A_N(x) := 1/|P_N| ∑_q∈ P_Nγ^q(x), P_N:={q≤ N:q is prime}. For every 1<p<∞, we prove a strong maximal inequality for (A_N)_N≥2 on L_p(𝒩) and that A_N(x) converges bilaterally almost uniformly for every x∈ L_p(𝒩). The proof exploits the circle method and a noncommutative sampling principle. For the convergence result, Bourgain's commutative argument uses pointwise maximal functions and exceptional sets. These tools are not available in the noncommutative setting. Instead, we show that the tails of the ergodic averages tend to zero in L_2(𝒩;ℓ_∞) and that the difference from the limit belongs to L_2(𝒩;c_0). This gives the desired b.a.u. convergence, and provides a positive answer to one question left open in \cite{ChenHongWang+arXiv2024}.
ER  -

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