In this paper we consider
primarily algebras F(T) of continuous funtions taking a topological space T into a
complete nonarchimedean nontrivially valued field F. Some general properties of
function algebras and topological algebras over valued fields are developed in §§1 and
2. Some principal results (Theorems 6 and 7) are analogs of theorems of Nachbin and
Shirota, and Warner: Essentially that F(T) with compact-open topology is
F-barreled iff unbounded functions exist on closed noncompact subsets of T;
and that full Fréchet algebras are realizable as function algebras F(,ℳ)
where ℳ denotes the space of nontrivial continuous homomorphisms of the
algebra.
