A characterization of Riesz spaces which are Riesz isomorphic to C( X) for some completely regular space X

Abstract

Let E be an Archimedean Riesz space possessing a weak unit e and let Ω be the collection of all Riesz homomorphisms ø from E onto ℝ such that ø( e)=1. The Gelfand mapping G : x→ x^ on E is defined by x^( ø) = ø( x) for all ø∈Ω. We endow Ω with the topology induced by E (i.e., the weakest topology such that each x^ is continuous on Ω). The principal ideal in E generated by e is denoted by I d ( e). The main theorem in this paper says that the following statements (A) and (B) are equivalent. (A) There exists a completely regular space X such that E is Riesz isomorphic to the space C( X) of all real continuous functions on X. (B) The following conditions for the Riesz space E hold: (1) E is Archimedean and has a weak unit e; (2) Ω separates the points of E; (3) E is uniformly complete; (4) G( I d ( e)) is norm dense in the space C b ( Ω) of all real bounded continuous functions on Ω; (5) E is 2-universally complete with carrier space Ω. Some other conditions are mentioned and an example is given to show that condition (5) is necessary for (B) ⇒(A).