Compact Homomorphisms of Regular Banach Algebras

Abstract

Let A be a complex commutative Banach algebra and let M-A be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points phi(1), phi(2). M-A, there is an element a is an element of A for which (a) over cap (phi(1)) = 1, (a) over cap (phi(2)) = 0 and parallel to a parallel to <= C, where (a) over cap is the Gelfand transform of a is an element of A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim