Surjectivity of the Gelfand map in the general setting

Establish surjectivity of the Gelfand map I_F: A → C(Max(A), F) for every topological field F and every F-algebra A in the category A_F defined by properties (Gelfand, compact spectrum of each element, and the pm-ring property that every prime ideal lies in a unique maximal ideal), thereby upgrading the adjunction between compact F-Tychonoff spaces and A_F to a duality.

Background

The paper constructs a dual adjunction between the category of compact F-Tychonoff spaces and the category A_F of F-algebras that satisfy key structural properties (Gelfand, compact spectra, pm-ring). In special cases—disconnected complete fields (generalizing Van der Put) and complete fields satisfying Stone–Weierstrass—the authors prove that the Gelfand map is surjective, yielding a duality.

Beyond these special cases, the adjunction’s upgrade to a full duality hinges entirely on the surjectivity of the Gelfand map I_F. The authors note that this surjectivity remains unresolved in the general setting, making it the primary barrier to a complete, field-agnostic duality.