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 c 0 , 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 → c 0 is an almost Dunford–Pettis operator.