Extreme maximal ideals of a partially ordered vector space

Abstract

1. Let V be a partially ordered vector space with an order unit e. It is well known that the class 9W of maximal ideals of V is in one-one correspondence with the class K of normalized positive linear functionals, in the sense that to each MC9Y corresponds a positive linear functional Om with M as its null-space and with OkM(e) = 1. A maximal ideal MC9Y is said to be extreme if OM is an extreme point of the convex set K; i.e., if the equation OM = Oq$+(l -u)O2 with 4i, q2CK and 0< o<1 can hold only with 01=42=q$M. Kadison [2 ] has fully demonstrated the importance of the extreme maximal ideals in the theory of the representation of partially ordered vector spaces by function spaces. The main purpose of this article is to give a simple and direct characterization of the extreme maximal ideals in terms of the order structure in V. This characterization improves our understanding of the place of the extreme maximal ideals in the representation theory. For example, it becomes almost obvious that, if V is lattice ordered, then the extreme maximal ideals coincide with the lattice maximal ideals. We consider a class of ideals that we call ideals. Besides providing the required characterization of the extreme maximal ideals as perfect maximal ideals, perfect ideals are of interest in their own right. The class of all perfect ideals has the most important properties of the class of all ideals. In particular, if V has no perfect proper ideals other than (0), then V is isomorphic to R. The fundamental theorem on extreme points of convex sets of linear functionals is the KreinMilman existence theorem [3 ]. This theorem is shown to be a simple consequence of the fact that each perfect proper ideal is contained in a perfect maximal ideal. In fact, the Krein-Milman theorem is related to this property of perfect ideals in the same way that the Hahn-Banach theorem is related to the corresponding property of ideals in general.