Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak⁎ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into c0, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E is almost limited if and only if every continuous linear operator T:E→c0 is an almost Dunford-Pettis operator.