Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. 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.