The Gel′fand subalgebra of real or non-Archimedean valued continuous functions

Abstract:

Let $L$ be either the field of real numbers or a nonarchimedean rank-one valued field. For topological space $T$ we study the Gelfand subalgebra ${C_0}(T,L)$ of the algebra of all $L$-valued continuous functions $C(T,L)$. The main result is that if $T$ is a paracompact locally compact Hausdorff space, which is ultraregular if $L$ is nonarchimedean, then for $f \in C(T,L)$ the following statements are equivalent: (1) There exists a compact set $K \subset T$ such that $f(T - K)$ is finite, (2) $f$ has finite range on every discrete closed subset of $T$, and (3) $f \in {C_0}(T,L)$.