Generalizing Gelfand duality to Nachbin spaces
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910001664098304 |
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| author | Bezhanishvili, G. Morandi, P. J. |
| author_facet | Bezhanishvili, G. Morandi, P. J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18807 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalizing Gelfand duality to Nachbin spaces Bezhanishvili, G. Morandi, P. J. Commutative Algebra Functional Analysis 06F25, 13J25, 54C30, 54E05, 54F05 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. |
| title | Generalizing Gelfand duality to Nachbin spaces |
| topic | Commutative Algebra Functional Analysis 06F25, 13J25, 54C30, 54E05, 54F05 |
| url | https://arxiv.org/abs/2601.18807 |