Generalizing Gelfand duality to Nachbin spaces
G. Bezhanishvili, P. J. Morandi
TL;DR
The paper generalizes Gelfand duality to the realm of Nachbin spaces by introducing Nachbin proximities on bal-algebras and proving a dual equivalence between $Nach$ and the category of uniformly complete bal-algebras with a closed Nachbin proximity. It also develops an alternative sbal-based route that yields De Rudder–Hansoul duality, and it establishes a pair of equivalent, dual frameworks (via $unba\ell$ and $usbal$) that connect Nachbin spaces to algebraic structures through generalized Stone–Weierstrass and Dieudonné-type results for Nachbin spaces. This work situates dualities for compact ordered spaces within a lattice-theoretic Proximity/Gelfand paradigm and unifies several threads in pointfree topology. The findings offer concrete tools for translating order-topological properties of Nachbin spaces into algebraic data and vice versa, with potential implications for ordered function spaces and noncommutative generalizations in the real-valued setting.
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 Dieudonné'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.