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 : L1(G) → A with dense range, where L1(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 $$A^{*} \cdot A = \overline{{{\text{span}}}} M_{A}$$ . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if $$A^{*} = \overline{{{\text{span}}}} M_{A}$$ . Furthermore, any of these conditions implies that M A is scattered. Several applications are given.