Nonnuclear subalgebras of AF algebras

We show that any non-type I separable unital AF algebra B can be modeled from inside by a nonnuclear C*-algebra and from outside by a nonexact C*-algebra. More precisely there exist unital separable quasidiagonal C*-algebras A ⊂ B ⊂ C of real rank zero, stable rank one, such that A is nonnuclear, C is nonexact, and both A and C are asymptotically homotopy equivalent to B. In particular A, B and C have the same ordered K-theory groups, hence isomorphic ideal lattices, and, A and B have (affinely) homeomorphic trace spaces.