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PreprintCC BY 4.0arXiv

A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences

  • Léonard Cadilhac
  • Christian Le Merdy
  • Safoura Zadeh
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Abstract

Let (M,τ) be a semifinite von Neumann algebra, let J be a trace-preserving Jordan isomorphism, and let (nk)k≥ 1 be a random increasing sequence of integers obtained by selecting each integer n≥ 1 independently with probability n−α, where 0<α<1/2. We show that, almost surely, for every x∈ L1(M), $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random L1 ergodic theorem to the non-commutative setting.

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Cite

BibTeXRIS
BibTeX
@misc{cadilhac2026noncommutative,
  title = {{A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences}},
  author = {Léonard Cadilhac and Christian Le Merdy and Safoura Zadeh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07970},
  eprint = {2609.07970},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07970},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences
AU  - Léonard Cadilhac
AU  - Christian Le Merdy
AU  - Safoura Zadeh
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07970
UR  - https://arxiv.org/abs/2609.07970
AB  - Let (M,τ) be a semifinite von Neumann algebra, let J be a trace-preserving Jordan isomorphism, and let (n_k)_k≥ 1 be a random increasing sequence of integers obtained by selecting each integer n≥ 1 independently with probability n^−α, where 0<α<1/2. We show that, almost surely, for every x∈ L^1(M), $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random L^1 ergodic theorem to the non-commutative setting.
ER  -