Automatic continuity of certain isomorphisms between regular banach function algebras

Abstract

Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T defined from A into B is said to be separating or disjointness preserving if f.g ≡ 0 implies Tf. Tg ≡ 0, for all f, g ≡ A. In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B. Finally, we show that linear isometrics between regular uniform algebras are separating. As corollaries, a classical theorem of Nagasawa ([19]) and the Banach-Stone theorem (both for regular uniform algebras) are easily inferred.