M-spaces with quasi-interior points

Abstract

In this paper we characterize those locally convex lattices which can be represented as dense sublatices containing 1 in a space C(X) and whose topologies can be recognized as topologies of uniform convergence on selections of compact subsets 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].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. Also, Kuller [4] obtained an algebraic representation for a complete locally convex lattice with topology T satisfying what he called condition (U) and observed a topological correspondence analogous to that obtained by Michael [5] for locally m-convex algebras: namely T is finer than the topology of uniform convergence on certain compact sets. We show in Theorem 1 that condition (U) for a locally convex lattice (not necessarily complete) is equivalent to being representable in our sense, so that T is in fact equal to the topology of uniform convergence on a collection of compact sets. We also show in Theorem 1 that this representability is equivalent to having a quasi-interior point and what we call a unit condition.Given such a representation, we prove (Theorem 2) that the carrier set X and the compact subsets are unique, but that the possible topologies on X form a closed interval between a weakest a and a strongest cu.In §2 we show (Theorem 3) that the completion of a space having a quasi-interior point and unit condition can be identified with C(X, a>) in the topology of uniform convergence on these compact sets.In §3 we contrast this topology with the topology of uniform convergence on all compact subsets.One might wish to compare this representation with the representation of Banach lattices having quasi-interior points as continuous extended real-valued functions considered by Lotz and Schaefer [6]. Stone-Weierstrass Embeddability.A positive element e in a locally convex lattice V is a quasi-interior point if the order ideal generated by e is dense in V. We will say that V satisfies the unit condition if for each lattice semi-norm p in a generating (not