A vector lattice topology and function space representation

Abstract

A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called "order partition spaces," an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact A', and conversely each such CC{X) is an order partition space. {Ce{X) denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of CC{X) for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.