Five months ago, Marcel de Jeu and I published an article
titled
Direct
limits in categories of normed vector lattices and Banach
lattices.
The paper is written in a self-contained manner, covering many known
results so that readers don’t need to consult external references while
reading.
This blog post aims to highlight what is new in the paper:
- Introduction of direct limits to the category of Banach lattices with contractive (almost) interval-preserving maps, say .
- A counterexample showing that the canonical construction of direct limits does not work in the category .
- A proof that order continuity is an invariant property under direct limits in .
- A result: the density of in a Banach function space implies order continuity of the space, where is a locally compact Hausdorff space whose compact subsets are all metrizable.
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 . A Banach function space is a Banach lattice consists of measurable functions on a measure space. 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 .