Surjective isometries of spaces of continuous functions

Isometries X --> X certainly do not have to be surjective, generally. When they are surjective, there can be powerful algebraic consequences-a surjective Isometry between real normed spaces, for example, must be an affine map Isometries must be surjective sometimes. If X is finite, for example, any isometry X --> X must be surjective; if X is a finite-dimensional normed space then any linear isometry must be surjective. Schikhof proved that for spherically complete fields F, all isometries F --> F are surjective if (and only if) F has a finite residue class held. Hence, for example, any isometry Q(p) --> Q(p) must be surjective. What about isometries of spaces of functions? There is a strong connection between linear Isometries H of spaces of functions-even with very different types of metrics-and the property f g = 0 double right arrow H f H g = 0. We call a map H with this property separating. (Other aliases include Lamperti operators, separation-preserving operators, disjoint operators, disjointness-preserving operators and d-homomorphisms.) For Banach spaces C(X) and C(Y) of real-valued continuous functions on the compact spaces X and Y, a surjective linear isometry H : C(X) --> C(Y) must be separating. And even though the norm is quite different, a linear isometry of L-p[0, 1] (or l(p)), 1 less than or equal to p less than or equal to infinity, p not equal 2, onto itself must also be separating ([3], pp. 170-175). Now suppose that X and Y are compact 0-dimensional Hausdorff spaces and the functions in C (X) and C(Y) take values in a non-Archimedean valued field F. We show In Sec. 4 that a linear separating isometry H : C (X) --> C (Y) is surjective if it satisfies a 'functional separation' condition that we call 'detaching' (Def. 3.1). What happens if we weaken linear to additive? We lose the 'detaching iff surjective' equivalence but Schikhof's theorem (the one cited above), comes to the rescue when F is spherically complete and has a finite residue class field, then (Theorems 6 1 and 6 2) 'detaching' additive separating isometries H : C (X) --> C(Y) must be surjective (and conversely). Some examples demonstrate the necessity of certain hypotheses.