On a commutative extension of a Banach algebra

Abstract

Introduction. Let be a commutative semi-simple Banach algebra and let A(A) be the set of nonzero multiplicative functionals on Denote by A' the strongly closed span of (A) and by A the Banach space adjoint of A'. Modifying a construction of R. Arens [1] we introduce a multiplication in A under which A becomes a commutative Banach algebra. is algebraically isomorphic to a subalgebra of A and the isomorphism is continuous. Indeed, we will henceforth need that the embedding of in A be topological. If has a weak bounded approximate identity, then the algebra A?n of multipliers of (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 is, in addition to the above assumptions, regular and Tauberian and if A(A) is discrete, then Am and A are topologically and algebraically isomorphic. The main result is the following: For with approximate identity in jA( ??), an element F of A is a multiplier of if and only if F belongs locally to at each point of A(A). The multiplier algebra of the group algebra L1(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.