Convex vector lattices and l-algebras

Abstract

AbstractFor A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit eA, we have the Yosida representation  as a Riesz space in D(XA), the lattice of extended real valued functions on the space of eA-maximal ideas. This note is about those A for which  is a convex subset of D(XA); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity eA. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large)