On closedness of convex sets in Banach lattices

TL;DR

This paper investigates when order closed convex sets in Banach lattices are also closed in certain topologies, introducing properties like DOCP and OSSP to characterize this equivalence.

Contribution

It introduces the properties DOCP and OSSP and characterizes when order closedness implies topological closedness in Banach lattices.

Findings

Order closed convex sets are topologically closed under certain conditions.

DOCP and OSSP are key properties for closedness equivalence.

Characterization of Banach lattices where order closedness matches topological closedness.

Abstract

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 $\sigma(X,X_n^\sim)$, where $X_n^\sim$ 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 $X_n^\sim$ contains a strictly positive element, every order closed convex set in $X$ is $\sigma(X,X_n^\sim)$-closed if and only if $X$ has $DOCP$ and either $X$ or $X_n^\sim$ 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 $\sigma(X,X_{uo}^\sim)$, where $X_{uo}^\sim$ is the unbounded order continuous dual of $X$.