Limits of vector lattices

TL;DR

This paper characterizes when certain vector lattices of continuous functions can be decomposed into inverse limits of carriers of order continuous functionals, extending classical results and developing a duality theory for these limits.

Contribution

It generalizes the decomposition of $C(K)$ spaces to $C(X)$ spaces on realcompact spaces using inverse limits and duality theory, and introduces new characterizations of perfect Dedekind complete vector lattices.

Findings

$C(K)$ spaces decompose as $ ext{ extlbrackdbl} ext{carrier of a maximal singular family} ext{ extrbrackdbl}$.

$C(X)$ is lattice isomorphic to the order dual of some vector lattice iff it is an inverse limit of carriers of order continuous functionals.

A Dedekind complete vector lattice is perfect iff it is an inverse limit of carriers of order continuous functionals.

Abstract

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 $\ell^\infty$-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.