On dominated extensions in function algebras

Abstract

The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of C ( K ) C(K) , a positive continuous function p on K and a closed set F ⊂ K F \subset K such that every measure in the annihilator of A vanishes on F, every function f ∈ C ( F ) f \in C(F) satisfying | f ( s ) | ⩽ p ( s ) ( s ∈ F ) |f(s)| \leqslant p(s)(s \in F) extends to a function f ~ ∈ A \tilde f \in A satisfying | f ~ ( z ) | ⩽ p ( z ) ( z ∈ K ) |\tilde f(z)| \leqslant p(z)(z \in K) . In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.