Generalizing Gelfand duality to Nachbin spaces

We introduce the notion of a Nachbin proximity on a bounded archimedean $\mathrm{\ell }$-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 $\mathrm{\ell }$-semialgebras (sbal-algebras), from which we derive De Rudder–Hansoul duality.