In this paper, we establish the one-sided maximal ergodic inequalities for a
large subclass of positive operators on noncommutative Lpβ-spaces for a fixed
1<p<β, 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 Lpβ([0,1]). Our study falls into neither the
category of positive contractions considered by Junge-Xu \cite{JUX07} nor the
class of power bounded positive invertible operators considered by
Hong-Liao-Wang \cite{HOLW18}. 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 \cite{JUL07},
still satisfy the maximal ergodic inequality. We also discuss some other
examples, showing sharp contrast to the classical situation.Comment: v6: minor changes, with more details of proof in Section