A non-Archimedean Stone-Banach theorem

Abstract:

If the spaces $C(T,R)$ and $C(S,R)$ of continuous functions on $S$ and $T$ are linearly isometric, then $T$ and $S$ are homeomorphic. By the classical Stone-Banach theorem the only linear isometries of $C(T,R)$ onto $C(S,R)$ are of the form $x \to a(x \circ h)$, where $h$ is a homeomorphism of $S$ onto $T$ and $a \in C(S,F)$ is of magnitude 1 for all $s$ in $S$. What happens if $R$ is replaced by a field with a valuation? In brief, the result fails. We discuss "how" by way of developing a necessary and sufficient condition for the theorem to hold, along with some examples to illustrate the point.