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A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if$|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the firstpart of this paper, we study some functional-analytic aspects ofuo-convergence. We prove that uo-convergence is stable under passing to andfrom regular sublattices. This fact leads to numerous applications presentedthroughout the paper. In particular, it allows us to improve several results in[26,27]. In the second part, we use uo-convergence to study convergence ofCes\`aro means in Banach lattices. In particular, we establish an intrinsicversion of Koml\'os' Theorem, which extends the main results of [35,16,31] in auniform way. We also develop a new and unified approach to Banach-Saksproperties and Banach-Saks operators based on uo-convergence. This approachyields, in particular, short direct proofs of several results in [21,24,25].

Uo-convergence and its applications to Ces\`aro means in Banach lattices