Noncommutative maximal ergodic inequalities associated with doubling conditions

This paper is devoted to the study of 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 \[ A_{n}x=\frac{1}{m(V^{n})}\int_{V^{n}}\alpha_{g}xdm(g),\quad x\in L_{p}(\mathcal{M}),n\in\mathbb{N},1\leq p\leq \infty \] is of weak type $(1,1)$ and of strong type $(p,p)$ for $1 < p<\infty$. Consequently, the sequence $(A_{n}x)_{n\geq 1}$ converges almost uniformly for $x\in L_{p}(\mathcal{M})$ for $1\leq p<\infty$. 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-Xu's setting. Our approach is based on the quantum probabilistic methods as well as the random walk theory.