The main result of this article is a structure theorem on separable complex Banach spaces E such that K(E) is an M-ideal in L(E). It is shown that if such a space has the approximation property, then it is isomorphic to an complemented subspace of a space with a shrinking unconditional finite dimensional decomposition. The proof blends topological techniques with Banach algebra methods. The assumption that E is a complex space was later shown to be unnecessary in an article of D. Li and the first-named author [Ann. Mat. Fourier 39, No.2, 361- 372 (1989)].