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arXiv · 2508.11188

Totally Disconnected (non-metric) Gelfand Duality

Sebastián Rodríguez·Xavier Caicedo
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Abstract

We characterize those algebras over a disconnected uniformly complete topological field which are representable as algebras of continuous functions on compact topological spaces, generalizing thus Gelfand duality for non-archimedean normed fields (Van der Put theorem). More generally, we establish for any topological field F a (dual) adjunction between the category of compact F-Tychonoff spaces and a natural category of commutative F-algebras, which becomes a duality for fields satisfying the Stone-Weierstrass theorem. To obtain these results we do not utilize analytic tools, but the canonical group uniformity of the field and intrinsic properties of the algebras.

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Sebastián Rodríguez, Xavier Caicedo. 2025-08-15. Totally Disconnected (non-metric) Gelfand Duality. https://arxiv.org/abs/2508.11188

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