Generalizing Gelfand Duality to Nachbin Spaces

We introduce the notion of a Nachbin proximity on a bounded archimedean ℓ-algebra (baℓ-algebra), and show that Gelfand duality lifts to yield a dual equivalence between the category of uniformly complete baℓ-algebras equipped with a closed Nachbin proximity and that of Nachbin spaces (compact ordered spaces). The key ingredients of the proof include appropriate generalizations of the Stone-Weierstrass theorem and Dieudonné’s lemma. We also develop an alternate approach by means of bounded archimedean ℓ-semialgebras (sbal-algebras), from which we derive De Rudder–Hansoul duality.