The notion of uniform equicontinuity in measure at zero for sequences of additive maps from a normed space into the space of measurable operators associated with a semifinite von Neumann algebra is discussed. It is shown that uniform equicontinuity in measure at zero on a dense subset implies the uniform equicontinuity in measure at zero on the entire space, which is then applied to derive some non-commutative ergodic theorems.

In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp−spaces, 1 < p < ∞, 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 Lp,q. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.Date: February 8, 2015. 2010 Mathematics Subject Classification. 47A35(primary), 46L52(secondary).

Abstract. We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting.0. Introduction. The study of almost everywhere convergence of the ergodic averages in the non-commutative setting was initiated by a number of authors among whom we mention Lance [5] and Yeadon [11]. Individual ergodic theorems have been established for algebras with states as well as for algebras equipped with a semifinite trace. The study of almost everywhere convergence of weighted and subsequential averages in von Neumann algebras is relatively new. So far, not much is known in this direction. Recently, a non-commutative analog of the classical Banach Principle, on convergence of sequences of measurable functions generated by a sequence of linear maps on L p -spaces, was established in [3]. It is expected that, as in the commutative case, this principle will be instrumental in obtaining various convergence results for the averages in non-commutative setting. In [8], an individual ergodic theorem for subsequences was proved, where the proof was based on application of the "commutative" Banach Principle. In this paper we use the ergodic theorem of Yeadon [11] together with the results of [3], adjusted to the bilateral almost uniform convergence, to show that the main result of [8] also holds in the vNa setting.

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