The Fine Structure of the Kasparov Groups III: Relative Quasidiagonality

Details The Fine Structure of the Kasparov Groups III: Relative Quasidiagonality The Fine Structure of the Kasparov Groups III: Relative Quasidiagonality

2001-07-12

arxiv.org/abs/math/0107090v1

Mathematics

Operator Algebras

34 pages

Scientific paper

In this paper we identify QD(A,B), the quasidiagonal classes in KK_1(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.

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