The Fine Structure Of The Kasparov Groups II: Relative Quasidiagonality

. Quasidiagonality was introduced by P.R. Halmos for operators and quickly generalized to C -algebras. D. Voiculescu asked how quasidiagonality (in its various forms) was related to topological invariants. N. Salinas systematically studied the topology of the Kasparov groups KK (A; B) and showed that this topology is related to relative quasidiagonality. In this paper we identify QD (A; B), the quasidiagonal elements in KK (A; B), in terms of K (A) and K (B), and we use these results in various applications. Here is our central result. Let N denote the bootstrap category. Theorem. Suppose that A 2 N and A is quasidiagonal relative to B. Then there is a natural isomorphism QD (A; B) = P ext 1 Z (K (A); K (B)) \Gamma1 : Thus for A 2 N relative quasidiagonality is indeed a topological invariant. We also settle a question raised by L.G. Brown on the relation between relative quasidiagonality and the kernel of the natural map ` : Ext 1 Z (K (A); K (B)) \Gamma! Ext ..