Uo-convergence and Its Applications to Cesàro Means in Banach Lattices

A net (xα) in a vector lattice X is said to uo-converge to x if |xα−x|∧u 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 uo-convergence 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, 28]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós’ Theorem, which extends the main results of [36, 16, 32] 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 [22, 25, 26].