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

TL;DR

This paper explores the Levi property in ordered Banach spaces, establishing new theorems, generalizing classical results, and discussing open problems related to convergence of increasing sequences.

Contribution

It introduces four new theorems about the Levi property, extends classical results to broader settings, and formulates open problems in the theory of ordered Banach spaces.

Findings

Levi property follows from minimal upper bound convergence.

Separable spaces with normal cone imply Levi property from increasing sequence bounds.

A generalized Dini's theorem provides conditions for compact operator spaces.

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.