发表机构
Harbin Institute of Technology; City University of Hong Kong（哈尔滨工业大学; 香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非交换冯·诺依曼代数上沿素数的遍历平均，利用圆法和非交换采样原理证明强极大不等式与双侧几乎一致收敛，解决了相关公开问题。
AI 中文摘要
设$\u27e8\uc120\u27e9$（即$\u2113$）为配备正规忠实半有限迹的冯·诺依曼代数，$\u03b3$为$\u2113$的保迹自同构。我们研究沿素数的遍历平均：$A_N(x):=\frac1{|P_N|}\u2211_{q\u2208P_N}\u03b3^q(x)$，其中$P_N:=\u201cq\u2264N:q\u00a0\text{is prime}\u201d$。对任意$1<p<\u221e$，我们证明了$(A_N)_{N\u22652}$在$L_p(\u2113)$上的强极大不等式，且对任意$x\u2208L_p(\u2113)$，$A_N(x)$双侧几乎一致收敛。证明利用了圆法和非交换采样原理。对于收敛结果，Bourgain的交换论证使用逐点极大函数和例外集，这些工具在非交换框架中不可用。我们转而证明遍历平均的尾部在$L_2(\u2113;\u2113_\u221e)$中趋于零，且与极限的差属于$L_2(\u2113;c_0)$。这得到了所需的b.a.u.收敛，并正面回答了ChenHongWang+arXiv2024中遗留的一个问题。
英文摘要
Let $\mathcal N$ be a von Neumann algebra equipped with a normal faithful semifinite trace, and let $γ$ be a trace-preserving automorphism of $\mathcal N$. We consider the ergodic averages along the prime numbers \[ A_N(x) := \frac1{|P_N|} \sum_{q\in P_N}γ^q(x), \qquad P_N:=\{q\leq N:q\ \text{is prime}\}. \] For every $1<p<\infty$, we prove a strong maximal inequality for $(A_N)_{N\geq2}$ on $L_p(\mathcal N)$ and that $A_N(x)$ converges bilaterally almost uniformly for every $x\in L_p(\mathcal N)$. 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 $L_2(\mathcal N;\ell_\infty)$ and that the difference from the limit belongs to $L_2(\mathcal N;c_0)$. This gives the desired b.a.u. convergence, and provides a positive answer to one question left open in \cite{ChenHongWang+arXiv2024}.
Comments32 pages
