Uo-convergence and its applications to Cesàro means in Banach lattices

A net (x α ) in a vector lattice X is said to uo-converge to x if $$\left| {{x_\alpha } - x} \right| \wedge u\xrightarrow{o}0$$ for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uoconvergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [27, 26]. In the second part, we use uo-convergence to study convergence of Cesaro means in Banach lattices. In particular, we establish an intrinsic version of Komlos’ Theorem, which extends the main results of [35, 16, 31] in a uniform way. We also develop a new and unified approach to Banach–Saks properties and Banach–Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [20, 24, 25].