Completely positive maps into corona algebras

We prove a decomposition theorem similar to the well-known result of Voiculescu's: namely that completely positive maps A -> M(B)/B factor through a given homomorphism tau: A -> M(B)/B when the homomorphism t has a certain infiniteness property. The algebra B is only assumed to be separable and nonunital; in particular, it is not assumed to be stable. The C*-algebra A is assumed to be separable, unital and nuclear.