Module Homomorphisms and Topological Centres Associated with Weakly Sequentially Complete Banach Algebras
✍ Scribed by John Baker; Anthony To-Ming Lau; John Pym
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 343 KB
- Volume
- 158
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
This paper is a contribution to the theory of weakly sequentially complete Banach algebras A. 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 algebras L 1 (G) for compact groups G or the Fourier algebras A(G) for discrete amenable groups G. In Section 2 we present our main result, that the topological centre (or set of weak* bicontinuous elements) of A** is identifiable with A. As a corollary, we deduce in Section 3 that each left A-module homomorphism from A* to A*A can be realized as right translation by an element of A. These conclusions generalise recent advances in the subject. In Section 4 we take a special algebra, l 1 (S ) for a commutative discrete semigroup S, and show that if its second dual has an identity then that identity must lie in l 1 (S).