Direct limits in categories of normed vector lattices and Banach lattices

Chun Ding, Marcel de Jeu

Published 2022-08-29Version 1

After collecting and establishing a number of results on interval and almost interval preserving linear maps, we show that direct systems in various categories of normed vector lattices and Banach lattices have direct limits which coincide with their direct limits in natural supercategories. For those categories where the general constructions do not work to establish the existence of general direct limits, we describe the basic structure of those direct limits that $\textit{do}$ exist. A direct system in the category of Banach lattices and contractive almost interval preserving vector lattice homomorphisms has a direct limit. When the Banach lattices in the system all have order continuous norms, then so does the Banach lattice in a direct limit. This is used to show that a Banach function space over a topological space has an order continuous norm when the topologies on all compact subsets are metrisable and (the images of) the continuous compactly supported functions are dense.