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).