Noncommutative maximal ergodic inequalities associated with doubling conditions

Abstract

We study noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $\alpha$ be a continuous action of $G$ on a von Neumann algebra $\mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by $$A_{n}x=\frac{1}{m\left(V^n\right)}{\int }_{V^n}{\alpha }_{g}xdm\left(g\right),x\in L_p\left(\mathcal{M}\right),n\in \mathbb{N},1\le p\le \infty ,$$ are of weak type $(1,1)$ and of strong type $(p,p)$ for $1<p<\infty$. Consequently, the sequence $(A_{n}x{)}_{n\ge 1}$ converges almost uniformly for $x\in L_p\left(\mathcal{M}\right)$ for $1\le p<\infty$. Also, we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions, and we prove the corresponding results for general actions on one fixed noncommutative $L_p$-space which are beyond the class of Dunford–Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy–Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford–Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge and Xu’s setting. Our approach is based on quantum probabilistic methods as well as random-walk theory.