  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.
