Increasing sequences in ordered Banach spaces -- new theorems and open problems

Abstract

An ordered Banach space $X$ is said to have the Levi property or to be regular if every increasing order bounded net (equivalently, sequence) is norm convergent. We prove four theorems related to this classical concept: (i) The Levi property follows from the - formally weaker - assumption that every increasing net that has a minimal upper bound is norm convergent. This motivates a discussion about in which sense the Levi property resembles the notion of order continuous norm from Banach lattice theory. (ii) If $X$ is separable and has normal cone, then the assumption that every increasing order bounded sequence has a supremum implies the Levi property. This generalizes a classical result about Banach lattices, but requires new ideas since one cannot work with disjoint sequences in the proof. (iii) A version of Dini's theorem for ordered Banach spaces that is more general than what is typically stated in the literature. We use this to derive a sufficient condition for the space of all compact operators between two Banach lattices to have the Levi property. (iv) Dini's theorem never holds on reflexive ordered Banach spaces with non-normal cone - i.e., on such a space one can always find an increasing sequence that converges weakly but not in norm. We illustrate our results by various examples and counterexamples and pose four open problems.