Amenability of Banach algebras of compact operators

Abstract

In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X).