The Fine Structure of the Kasparov Groups III: Relative Quasidiagonality

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In this paper we identify QD(A,B), the quasidiagonal classes in KK₁(A,B), in terms of K*(A) and K*(B), and we use these results in various applications. Here is our central result. Theorem: Suppose that A is in the category of separable nuclear C*-algebras which satisfy the UCT and A is quasidiagonal relative to B. Then there is a natural isomorphism QD(A,B) = Pext (K*(A), K*(B))₀. Thus quasidiagonality of KK-classes is indeed a topological invariant. We give several applications. Finally, we establish a converse to a theorem of Davidson, Herrero, and Salinas, giving conditions under which the quasidiagonality of A/K implies the quasidiagonality of the associated representation of A.