On dual and three space problems for the compact approximation property

We introduce the properties W∗D and BW∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X∗ has the CAP and the W∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M⊥ is complemented in X∗ and X∗ has the W∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.