Module Homomorphisms and Topological Centres Associated with Weakly Sequentially Complete Banach Algebras

This paper is a contribution to the theory of weakly sequentially complete Banach algebrasA. We require them to have bounded approximate identities and, for the most part, to be ideals in their second duals, so that examples are the group algebrasL1(G) for compact groupsGor the Fourier algebrasA(G) for discrete amenable groupsG. In Section 2 we present our main result, that the topological centre (or set of weak* bicontinuous elements) ofA** is identifiable withA. As a corollary, we deduce in Section 3 that each leftA-module homomorphism fromA* toA*Acan be realized as right translation by an element ofA. These conclusions generalise recent advances in the subject. In Section 4 we take a special algebra,l1(S) for a commutative discrete semigroupS, and show that if its second dual has an identity then that identity must lie inl1(S).