Direct limits in categories of normed vector lattices and Banach lattices (Q6111061)

This paper deals with existence and computation of direct (or inductive or co-)limits in categories of normed vector lattices and Banach lattices. On the one hand, both questions are of great importance when abstract category methods ought to be applied. On the other hand, as the categories under consideration combine algebraic structures (vector space + lattice) with analytic structures (norm topology), it is to be expected that existence of arbitrary direct limits may in general fail and that it might be hard to compute such limits explicitly. In the current paper, the authors give a comprehensive account to the aforementioned questions. Indeed, they study thirteen different categories of vector lattices, normed vector lattices and Banach lattices by furnishing the latter with different classes of morphisms, e.g., interval preserving lattice homomorphisms, contractive lattice homomorphisms etc. Sections~2 and~3 of the paper contain a detailed account of the aforementioned categories summarizing the basic theory. Section~4 and~5 give three standard constructions of direct limits of vector spaces and normed spaces with contractions as morphisms. The authors identify here in which cases the standard constructions still work if one considers in addition to the structure of a vector space or normed space also a lattice structure on the steps of the direct limit. One of the major results of Section~4 is that a direct system in the category of Banach lattices with contractive almost interval preserving vector lattice homomorphisms has a direct limit. The authors also give examples illustrating why in other cases the standard construction fails; they further mention that for six of the thirteen cateogories the existence in full generality is still open. Section~6 concludes the paper with results on the order continuity of direct limits of Banach lattices.