A Characterization of Commutative Group Algebras and Measure Algebras

Abstract

MARC A. RIEFFEL(')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 [ll] on the concrete representation of abstract real L-spaces, we show that any abstract complex L-space is isometrically isomorphic to complex Ll{X,m) for some measure space (X, m).Let B be any Banach space and / any nonzero linear functional on B. Let P(f) = \x: f{x) = \\f\\ \\x\\ \.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 / be a multiplicative linear functional (m.l.f.) on A. We say that /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 that if / is any m.l.f. of L(G) or M(G) corresponding to a (continuous) character of G, then / is L'-inducing.Our characterization of group algebras is a partial converse of this observation: Theorem A. Let A be a commutative semisimple Banach algebra such that (1) Every m.l.f. of A is L'-inducing,(2) A is Tauberian.