Abstract:
After collecting a number of results on interval and almost interval preserving linear
maps and vector lattice homomorphisms, we show that direct systems in various
categories of normed vector lattices and Banach lattices have direct limits, and that
these coincide with direct limits of the systems in naturally associated other categories.
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 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 locally compact Hausdorff 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.
