  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.
