Generalizing Gelfand duality to Nachbin spaces
Commutative Algebra
2026-01-28 v1 Functional Analysis
Abstract
We introduce the notion of a Nachbin proximity on a bounded archimedean -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 -semialgebras (sbal-algebras), from which we derive De Rudder--Hansoul duality.
Cite
@article{arxiv.2601.18807,
title = {Generalizing Gelfand duality to Nachbin spaces},
author = {G. Bezhanishvili and P. J. Morandi},
journal= {arXiv preprint arXiv:2601.18807},
year = {2026}
}