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

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.