On a Commutative Extension of a Banach Algebra1

Introduction. Let A be a commutative semi-simple Banach algebra and let A(.4) be the set of nonzero multiplicative functionals on A. Denote by A' the strongly closed span of A(^4) and by A" the Banach space adjoint of A'. Modifying a construction of R. Arens [l] we introduce a multiplication in A" under which A" becomes a commutative Banach algebra. A is algebraically isomorphic to a subalgebra of A" and the isomorphism is continuous. Indeed, we will henceforth need that the embedding of A in A" be topological. If A has a weak bounded approximate identity, then the algebra Am of multipliers of A (see [3; 4]) is likewise embeddable in A" and the isomorphism is again continuous. In this paper we are concerned with identification of Am in A". For example, if A is, in addition to the above assumptions, regular and Tauberian and if A(^4) is discrete, then Am and A" are topologically and algebraically isomorphic. The main result is the following: For A with approximate identity in Ja( °°), an element £of A" is a multiplier of A if and only if £ belongs locally to A at each point of A(.4). The multiplier algebra of the group algebra LX(G) of a locally compact abelian group G is the algebra M(G) of bounded measures on G. In this case, our main theorem closely parallels Eberlein's necessary and sufficient condition for a function to be a Fourier-Stieltjes transform of a measure on G. See [5]. As an application, we construct A" and use Eberlein's theorem to determine the algebra of multipliers of the Li-algebra of certain semi-groups G+. The author wishes to thank Professor I. Glicksberg for calling his attention to A" and for many helpful suggestions.