Compact homomorphisms of regular banach algebras

Abstract

Let A be a complex commutative Banach algebra and let MA 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 φ1 , φ2 ∈ MA , there is an element a ∈ A for whichâ (φ1 ) = 1,â (φ2 ) = 0 and a ≤ C, whereâ is the Gelfand transform of a ∈ 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.