(almost) interval preserving maps preserve order continuity

In the category of Banach lattices and (almost) interval preserving linear maps, a direct limit of a directed system of order continuous Banach lattices is again order continuous.

The above is an interesting result in the paper Direct limits in categories of normed vector lattices and Banach lattices joint with Marcel. For citation, you may copy the following:

@article{ding2023direct,
  title={Direct limits in categories of normed vector lattices and Banach lattices},
  author={Ding, Chun and de Jeu, Marcel},
  journal={Positivity},
  volume={27},
  number={3},
  pages={39},
  year={2023},
  publisher={Springer}
}

An immediate corollary:

You may prove this corollary without using our result on direct limits. The view is that our result points out that the order continuity is preserved by (almost) interval preserving linear maps, instead of lattice homomorphisms. Indeed, there is an example of a direct system of order continuous Banach lattices whose direct limit is not order continuous in the category of Banach lattices and lattice homomorphisms.

Quick review of definitions

A Banach lattice is a Banach space with an lattice order such that such that for all $x,y\in X$, the implication $${|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}$$ holds, where the absolute value $|\cdot|$ is defined as ${|x|=x\vee (-x)}$. Canonical examples include $L^p(X)$ and $C_0(X)$ where $X$ is a locally compact Hausdorff space.

We call a Banach lattice order continuous if every monotone order bounded sequence is convergent in norm.

A linear map $\map\colon E\to F$ between Banach lattices is said to be interval preserving if it is positive and such that $\map([0,x]_E)=[0,\map(x)]_F$ for all positive $x$ in $E$.

A linear map $\map\colon E\to F$ between Banach lattices is called almost interval preserving if it is positive and such that $[0, \map(x)]_F=\overline{\map([0,x]_E)}$ for all all positive $x$ in $E$.