Totally Disconnected (non-metric) Gelfand Duality

TL;DR

This paper generalizes Gelfand duality to a broad class of topological fields, characterizing algebras of continuous functions on compact spaces without relying on analytic methods.

Contribution

It establishes a duality between categories of compact F-Tychonoff spaces and commutative F-algebras for any topological field F, extending classical Gelfand duality.

Findings

Characterizes algebras over disconnected fields as function algebras

Establishes a dual adjunction for any topological field F

Achieves duality under the Stone-Weierstrass theorem

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.