  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).
