A characterization of commutative group algebras and measure algebras

Abstract

Let G be a locally compact Abelian group. By the group algebra, L(G), of G we mean the Banach algebra of all (equivalence classes of) complexvalued functions on G integrable with respect to the Haar measure on G, with convolution as multiplication. By the measure algebra, M(G), of G we mean the Banach algebra of all finite complex-valued regular Borel measures on G, with convolution as multiplication. In this paper we give a characterization of those commutative Banach algebras which are the group algebras of locally compact Abelian groups, and we give a characterization of those commutative Banach algebras which are the measure algebras of locally compact Abelian groups. Our results have been announced already, without proofs, in [13]. We will now give a brief outline of our results. In ?1 we define an abstract complex L-space, and, in extension of Kakutani's theorem [11] on the concrete representation of abstract real L-spaces, we show that any abstract complex L-space is isometrically isomorphic to complex L'(X, m) for some measure space (X, m). Let B be any Banach space and f any nonzero linear functional on B. Let P(f) = {x: f(x) = l/fIl lxix }. Then P(f) is a cone in B and so defines a partial order on B in the usual way. In ?2 we study this phenomenon briefly, with particular attention to the case in which B under this order becomes an abstract complex L-space. Now let A be any complex Banach algebra, with multiplication denoted by *, and let f be a multiplicative linear functional (m.l.f.) on A. We say that f is L'-inducing if, under the order defined by P(f ), A is an abstract complex L-space, and if two other hypotheses are satisfied. In ?3 we see