Noncommutative extensions of an ergodic theorem of Breiman

Abstract

Let $(\mathcal{M},\tau)$ be a tracial von Neumann algebra, $T:L_1(\mathcal{M},\tau)\to L_1(\mathcal{M},\tau)$ be induced by a normal $\tau$-preserving $*$-automorphism of $\mathcal{M}$, and $x\in L_1(\mathcal{M},\tau)$. We will prove the bilaterally almost uniform convergence of averages of the form $$\frac{1}{n}\sum_{k=0}^{n-1}T^k(x_k),$$ generalizing an ergodic theorem proven by Breiman in \cite{Breiman57}, where $(x_k)_{k=0}^{\infty}$ is an element of the noncommutative vector-valued $L_p$-space $L_p(\mathcal{M},\tau;\ell_\infty)$ such that there exists $x\in L_p(\mathcal{M},\tau)$ with $(x_k-x)_{k=0}^{\infty}\in L_p(\mathcal{M},\tau;c_0)$. A generalization to semifinite $(\mathcal{M},\tau)$ and general positive Dunford-Schwartz operators $T\in DS^+(\mathcal{M},\tau)$ is shown as well when $1<p<\infty$. In addition to these, we also prove a number of generalizations of the results to weighted, subsequential, and other types of averages.