Bounded point derivations on certain Banach algebras

Abstract

Let now X be a compact set in the z-plane. Denote by R,(X) the algebra of all rational functions having no poles on X. Let R(X) be the closure of R,(X) in the uniform norm on X. R(X) is then a Banach algebra in this norm. It is well known that the homomorphisms of R(X) into the scalars are all of the formf-+f( x ), w h ere x is some point in X. We identify the point x with the homomorphism it induces. Fix x in X. It is easy to verify that there exists a bounded point derivation at x if and only if