Noncommutative maximal ergodic inequalities associated with doubling conditions

Abstract

In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong-Liao-Wang. Our strategy essentially relies on various structural characterizations and dilation properties associated with Lamperti operators, which are of independent interest. More precisely, we give a structural description of Lamperti operators in the noncommutative setting, and obtain a simultaneous dilation theorem for Lamperti contractions. As a consequence we establish the maximal ergodic theorem for the strong closure of the convex hull of corresponding family of positive contractions. Moreover, in conjunction with a newly-built structural theorem, we also obtain the maximal ergodic inequalities for positive power bounded doubly Lamperti operators. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge-Le Merdy, still satisfy the maximal ergodic inequalities. We also discuss some other examples, showing sharp contrast to the classical situation.