Amenable and weakly amenable Banach algebras with compact multiplication

We investigate amenable and weakly amenable Banach algebras with compact mu ltiplication. Any amenable Banach algebra with compact multiplication is bi projective. As a consequence, every semisimple such algebra which has appro ximation property is a topological direct sum of full matrix algebras. In t he radical case no such structure theorem is at hand. We also investigate B anach algebras which have a bounded approximate identity consisting of norm alized 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 c onstruct a radical example with compact multiplication which moreover is an integral domain. This furnishes a new example of a commutative, weakly ame nable, non-amenable, radical Banach algebra. (C) 2000 Academic Press.