Additive bijections of C(X)

For compact Hausdorff spaces X and Y, the Stone–Banach theorem asserts that surjective linear isometries H:C(X)→C(Y) are of the form Hf(y)=f(h(y))H1(y) (f∈C(X), y∈Y and 1(x)≡1) where h:Y→X is a homeomorphism and |H1(y)|≡1. Omitting the requirements that h be a homeomorphism and that |H1(y)|≡1, maps of this type f↦(f∘h)H1 are called `weighted composition mapsʹ where H1∈C(Y) is the `weightʹ function. Instead of K=R or C, suppose (K,| |) is a valued field. We now consider K-valued continuous functions C(X,K) and C(Y,K). Now linear isometries H:C(X,K)→C(Y,K) may take different forms. Indeed, if (K,| |) is non-Archimedean (i.e., |a+b|≤max(|a|,|b|)), a linear isometry H:C(X,K)→C(Y,K) is a weighted composition if and only if it is separating in the sense that, for all f,g∈C(X,K), fg=0⇒HfHg=0. We weaken linear to additive and isometry to separating bijection and consider what forms such a bijection H:C(X,K)→C(Y,K) can have for K=R, C or a non-Archimedean valued field. We show in Theorem 18 that an additive separating bijection H:C(X,K)→C(Y,K) is automatically continuous; it is a weighted composition map with a homeomorphism if K=R or Qp the p-adic numbers) and `almostʹ a weighted composition if K=C (see Theorem 18(b)).