Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911627849236480 |
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| author | Merdy, Christian Le Zadeh, Safoura |
| author_facet | Merdy, Christian Le Zadeh, Safoura |
| contents | 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 $\mathbb{P}(X_n = 1) = n^{-α}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces Merdy, Christian Le Zadeh, Safoura Operator Algebras Dynamical Systems Probability 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 $\mathbb{P}(X_n = 1) = n^{-α}$, 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. |
| title | Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces |
| topic | Operator Algebras Dynamical Systems Probability |
| url | https://arxiv.org/abs/2604.25029 |