2018/08/21 by Tantrawan, Made, Leung, Denny H.
#46A20 #46A55 #46B42 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1808.06747
Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology σ(X,Xn^∼), where Xn^∼ is the order continuous dual of X. Motivated by the solution in the Orlicz space case, we introduce two relevant properties: the disjoint order continuity property (DOCP) and the order subsequence splitting property (OSSP). We show that when X is monotonically complete with OSSP and Xn^∼ contains a strictly positive element, every order closed convex set in X is σ(X,Xn^∼)-closed if and only if X has DOCP and either X or Xn^∼ is order continuous. This in turn occurs if and only if either X or the norm dual X^* of X is order continuous. We also give a modular condition under which a Banach lattice has OSSP. In addition, we also give a characterization of X for which order closedness of a convex set in X is equivalent to closedness with respect to the topology σ(X,Xuo^∼), where Xuo^∼ is the unbounded order continuous dual of X.