Noncommutative maximal inequalities for polynomial ergodic averages

Abstract

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.