Limits of vector lattices

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 ℓ∞-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.