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arXiv · 2601.18807

Generalizing Gelfand duality to Nachbin spaces

G. Bezhanishvili·P. J. Morandi
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Abstract

We introduce the notion of a Nachbin proximity on a bounded archimedean $\ell$-algebra (bal-algebra). We prove that Gelfand duality lifts to yield a dual equivalence between the categories of uniformly complete bal-algebras equipped with a closed Nachbin proximity and 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.

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G. Bezhanishvili, P. J. Morandi. 2026-01-15. Generalizing Gelfand duality to Nachbin spaces. https://arxiv.org/abs/2601.18807

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