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:

Let be a Banach lattice, and let be a family of closed ideals in . If each is order continuous, so is .

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
, the
implication
holds, where the absolute value
is defined as
.
Canonical examples include
and
where
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
between Banach lattices is said to be *interval preserving* if it
is positive and such that
for all positive
in
.

A linear map
between Banach lattices is called *almost interval preserving* if
it is positive and such that
for all all positive
in
.