A Vector Lattice Topology and Function Space Representation

Abstract

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.A classical theorem of Kakutani (see [10]) states that each order unit space with its order unit topology is lattice isometric to a dense subspace of C(X) containing the constant functions with supremum norm for some compact X and, conversely, each such C(X) is an order unit space. (Here C(X) denotes the space of real-valued continuous functions on X.) In § 1 we generalize the concept of an order unit and the topology which it generates and provide an extension of Kakutani's theorem for this setting. Specifically, we consider a vector lattice having a weak order unit e and a partition of e which, in analogy to the order unit case, defines a locally convex topology. Such a space, called an "order partition space," is shown (Theorem 1) to be lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact X. (Here CC(X) denotes C(X) with the topology of compact convergence.) Conversely, CC(X) for X locally compact is an order partition space. The authors obtained a similar result in Theorem 1 of [5] for the restricted case that X is a countable union of open compact sets. In the remainder of §1, sufficient conditions are discussed for the order partition topology to be lattice-theoretic-i.e., independent of the choice of e

The order topology for function lattices and realcompactness

“…We map V into C(X) by the usual Gelfand map (x) x(v) for all x in X. We will refer to X as the carrier space of V. The proof of the following proposition uses the techniques of Lemma 2 in [3].…”

M-spaces with quasi-interior points

“…Here C(X) is the lattice of continuous real-valued functions on a completely regular space X. The class of such locally convex lattices includes the classical order unit spaces investigated by Kakutani [3], arbitrary products of order unit spaces, for example [I L", and the order partition spaces studied in [1].We remark that Jameson [2] obtained a representation theorem for arbitrary Mspaces as sublattices of C(X) with topologies of uniform convergence on certain compact subsets. In his general setting the sublattices need not be dense nor separate the points of X.…”

“…Here C(X) is the lattice of continuous real-valued functions on a completely regular space X. The class of such locally convex lattices includes the classical order unit spaces investigated by Kakutani [3], arbitrary products of order unit spaces, for example [I L", and the order partition spaces studied in [1].…”

“…More generally, it would be of interest to know conditions for an M*-space to be represented with the topology of uniform convergence on all compact subsets of (X, w). The orthogonality condition studied in [1] provides one such condition in a restricted setting.…”