Separating maps on Fréchet algebras

Abstract

Let A be a normal Fréchet function algebra on a topological space X which is the projective limit of a sequence of certain Banach function algebras and satisfies Ditkin's condition. Let B be a function algebra on a topological space Y. We show that if T : A → B is a separating map, that is f.g = 0 implies T f.T g = 0, then T can be represented as T f (y) = T 1(y)f (h(y)), for each f ∊ A, on a subset of Y. Using this result we show that if B is also a Fréchet function algebra and T is a separating bijection then it is automatically continuous.