Amenable and Weakly Amenable Banach Algebras with Compact Multiplication

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.