Unbounded order convergence in dual spaces

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 uo-convergent 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.