Pointwise convergence of noncommutative ergodic averages along the primes

Let $\mathcal{N}$ be a von Neumann algebra equipped with a normal faithful semifinite trace, and let $\gamma$ be a trace-preserving automorphism of $\mathcal{N}$. We consider the ergodic averages along the prime numbers $$A_N(x):=\frac{1}{\mathrm{\mid }{P}_N\mathrm{\mid }}\sum _{q\in P_N}{\gamma }^q(x),P_N:=\{q\le N:q\text{ is prime}\}\mathrm{.}$$ For every $1<p<\mathrm{\infty }$, we prove a strong maximal inequality for $(A_N{)}_{N\ge 2}$ 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 $L_2(\mathcal{N};{\mathrm{\ell }}_{\mathrm{\infty }})$ 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}.