A Class of Banach Algebras Whose Duals Have the Radon-Nikodym Property

Abstract

Let A be a complex, commutative Banach algebra and let M-A be the structure space of A. Assume that there exists a continuous homomorphism h : L-1(G) -> A with dense range, where L-1(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M-A is scattered (i.e., it has no nonempty perfect subset) and A* center dot A = (SPAN) over barM(A). (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if A* = (span) over bar M-A. Furthermore, any of these conditions implies that M-A is scattered. Several applications are given.