Limits of vector lattices

If K is a compact Hausdorff space so that the Banach lattice C(K) is isometrically lattice isomorphic to a dual of some Banach lattice, then C(K) can be decomposed as the 8 infinity-direct sum of the carriers of a maximal singular family of order continuous functionals on C(K). In order to generalise this result to the vector lattice C(X) of continuous, real valued functions on a realcompact space X, we consider direct and inverse limits in suitable categories of vector lattices. We develop a duality theory for such limits and apply this theory to show that C(X) is lattice isomorphic to the order dual of some vector lattice F if and only if C(X) can be decomposed as the inverse limit of the carriers of all order continuous functionals on C(X). In fact, we obtain a more general result: A Dedekind complete vector lattice E is perfect if and only if it is lattice isomorphic to the inverse limit of the carriers of a suitable family of order continuous functionals on E. A number of other applications are presented, including a decomposition theorem for order dual spaces in terms of spaces of Radon measures.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).

Automated summary

This paper investigates the isomorphism relationship between Banach lattice and dual lattice on compact Hausdorff spaces, and presents some decomposition theorems and duality theories. The applications of direct and inverse limits in the category of vector lattices are also discussed.