Amenable and weakly amenable Banach algebras with compact multiplication

Research Report MRR99-011

Amenable and weakly amenable Banach algebras with compact multiplication

R. J. Loy, C. J. Read, V. Runde and G. A. Willis

Abstract: We investigate amenable and weakly amenable Banach algebras with compact multiplication. Any amenable Banach algebra with compact multiplication is biprojective. As a consequence, every semisimple such algebra which has the approximation property is a topological direct sum of full matrix algebras. In the radical case no such structure theorem is at hand. We also investigate Banach algebras which have a bounded approximate identity consisting of normalized powers of an element x. Any such Banach algebra is either unital or radical; if the algebra is also generated by x, it is weakly amenable. We construct a radical example with compact multiplication which moreover is an integral domain. This furnishes a new example of a commutative, weakly amenable, non-amenable, radical Banach algebra.

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