Totally Disconnected (non-metric) Gelfand Duality
Sebastián Rodríguez, Xavier Caicedo
TL;DR
The paper develops a purely algebraic framework to realize Gelfand duality for algebras over topological fields, including disconnected and non-archimedean cases. It introduces generalized Gelfand-Kolmogorov transforms and two functors, CF and MF, to connect compact F-Tychonoff spaces with commutative F-algebras, and defines a Gelfand topology and a canonical uniformity on spectra. A central result extends Van der Put's theorem to complete disconnected fields, showing that certain Gelfand algebras are exactly algebras of F-valued continuous functions on Stone spaces, and constructs a dual adjunction between compact F-Tychonoff spaces and suitable F-algebras. Under Stone-Weierstrass and completeness, this adjunction upgrades to a duality, providing an algebraic characterization of algebras of continuous functions across a broad class of topological fields. Collectively, the work yields a robust, intrinsic algebraic route to Gelfand duality beyond the classical complex setting, applicable to non-archimedean and disconnected fields.
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.