Ergodic theorems in fully symmetric spaces of $\tau-$measurable operators

TL;DR

This paper extends ergodic theorems and maximal inequalities from noncommutative Lp-spaces to fully symmetric Banach spaces, including Lorentz spaces, using noncommutative interpolation techniques and analyzing norm convergence of ergodic averages.

Contribution

It introduces new maximal ergodic inequalities in noncommutative fully symmetric spaces and applies them to establish ergodic theorems in these broader contexts.

Findings

Maximal ergodic inequalities derived for noncommutative Lp-spaces.

Individual and weighted ergodic theorems extended to noncommutative symmetric spaces.

Norm convergence of ergodic averages established in these spaces.

Abstract

In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp-spaces, 1 < p < infinity, was established and, among other things, corresponding maximal ergodic inequalities and individual ergodic theorems were derived. In this article, we derive maximal ergodic inequalities in noncommutative Lp-spaces directly from [25] and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in particular, to noncommutative Lorentz spaces Lpq. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.