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

Abstract

AbstractIn this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.