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].

A net (x α ) in a Banach lattice X is said to un-converge to a vector x if |x α − x| ∧ u → 0 for every u ∈ X + . In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B X is un-complete. For a Banach lattice X, B X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.

Abstract. A net (x α ) in a vector lattice X is unbounded order convergent to x ∈ X if |x α − x| ∧ u converges to 0 in order for all u ∈ X + . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net (x α ) in a Banach lattice X is unbounded norm convergent to x if |x α − x| ∧ u → 0 for all u ∈ X + . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E n ) on a Banach lattice F is said to be a filtration ifthe Banach space of all norm uniformly bounded martingales. It is shown that if F doesn't contain a copy of c 0 or if every E n is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F → M → F * * . Finally, it is shown that every martingale difference sequence is a monotone basic sequence.

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