Generalizing Gelfand duality to Nachbin spaces

TL;DR

This paper extends Gelfand duality to Nachbin spaces by introducing Nachbin proximities on bal-algebras, establishing a dual equivalence between algebraic and topological categories with new generalizations of classical theorems.

Contribution

It generalizes Gelfand duality to the setting of Nachbin spaces using Nachbin proximities on bal-algebras, and develops an alternative approach via sbal-algebras.

Findings

Established duality between bal-algebras with Nachbin proximity and Nachbin spaces.

Generalized Stone-Weierstrass theorem for this setting.

Derived De Rudder--Hansoul duality through an alternative approach.

Abstract

We introduce the notion of a Nachbin proximity on a bounded archimedean $\ell$-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 Dieudonn\'{e}'s lemma. We also develop an alternate approach by means of bounded archimedean $\ell$-semialgebras (sbal-algebras), from which we derive De Rudder--Hansoul duality.