Characterization of topological fields satisfying the Stone–Weierstrass theorem

Characterize all topological fields F for which the Stone–Weierstrass theorem holds; namely, determine precisely those F such that for every compact Hausdorff space X, any subalgebra A ⊂ C(X, F) containing the constant functions and separating points is uniformly dense with respect to the canonical uniformity (and, in the complex case, A is additionally closed under complex conjugation).

Background

The Stone–Weierstrass theorem plays a crucial role in ensuring the surjectivity of the Gelfand map and hence in obtaining duality beyond the disconnected case. The authors highlight that the theorem holds for many fields, including those whose topology comes from an absolute value or a Krull valuation.

Despite these examples, a complete intrinsic characterization of topological fields that satisfy the Stone–Weierstrass theorem is not currently known, leaving a gap in understanding the scope of the general duality framework.