Direct limits in categories of normed vector lattices and Banach lattices

TL;DR

This paper investigates the existence and structure of direct limits in categories of normed and Banach lattices, revealing conditions under which these limits exist and their properties, especially in relation to order continuity and function spaces.

Contribution

It establishes the existence of direct limits in various categories of normed and Banach lattices and describes their structure, especially for systems with order continuous norms.

Findings

Direct limits exist in categories of Banach lattices with contractive almost interval preserving homomorphisms.

In systems with order continuous norms, the direct limit also has an order continuous norm.

Banach function spaces over certain locally compact spaces have order continuous norms under specified conditions.

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.