Abstract. A net (x α ) in a vector lattice X is said to uo-converge to x if |x α −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 [26,27]. 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 [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 [21,24,25].

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A net (x α ) in a vector lattice X is said to be unbounded order convergent (or uo-convergent, for short) to x ∈ X if the net (|x α − x| ∧ y) converges to 0 in order for all y ∈ X + . In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X be a Banach lattice. We prove that every norm bounded uoconvergent net in X * is w * -convergent iff X has order continuous norm, and that every w * -convergent net in X * is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ-order complete Banach lattices the spaces in whose dual space every simultaneously uo-and w * -convergent sequence converges weakly/in norm.

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