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In this paper, we establish the one-sided maximal ergodic inequalities for alarge subclass of positive operators on noncommutative $L_p$-spaces for a fixed$1<p<\infty$, which particularly applies to positive isometries and generalpositive Lamperti contractions or power bounded doubly Lamperti operators;moreover, it is known that this subclass recovers all positive contractions onthe classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither thecategory of positive contractions considered by Junge-Xu nor the class of powerbounded positive invertible operators considered by Hong-Liao-Wang. Ourstrategy essentially relies on various structural characterizations anddilation properties associated with Lamperti operators, which are ofindependent interest. More precisely, we give a structural description ofLamperti operators in the noncommutative setting, and obtain a simultaneousdilation theorem for Lamperti contractions. As a consequence we establish themaximal ergodic theorem for the strong closure of the convex hull ofcorresponding family of positive contractions. Moreover, in conjunction with anewly-built structural theorem, we also obtain the maximal ergodic inequalitiesfor positive power bounded doubly Lamperti operators.  We also observe that the concrete examples of positive contractions withoutAkcoglu's dilation, which were constructed by Junge-Le Merdy, still satisfy themaximal ergodic inequalities. We also discuss some other examples, showingsharp contrast to the classical situation.

Maximal ergodic inequalities for some positive operators on noncommutative $L_p$-spaces