Unbounded order convergence in dual spaces

A net (xα)(xα) in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈Xx∈X if the net (|xα−x|∧y)(|xα−x|∧y) converges to 0 in order for all y∈X+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⁎X⁎ is w⁎w⁎-convergent iff X   has order continuous norm, and that every w⁎w⁎-convergent net in X⁎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⁎w⁎-convergent sequence converges weakly/in norm.