Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 91-117.

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

Authors: Claude L. Schochet
Author institution: Mathematics Department, Wayne State University, Detroit, MI 48202, USA

Summary: 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: Let

$\widetilde{\mathcal N}$ denote the category of separable nuclear

$C^*$ -algebras which satisfy the Universal Coefficient Theorem. Suppose that

$A \in \widetilde{\mathcal N}$ and

$A$ is quasidiagonal relative to

$B$ . Then there is a natural isomorphism

$\begin{equation*} QD(A,B) \,\cong\, \mathrm{Pext}_{\mathbb Z}^1( {K_*(A)},{K_*(B)}) _{0} .\end{equation*}$ Thus, for

$A \in \widetilde{\mathcal N}$ quasidiagonality of

$KK$ -classes is indeed a topological invariant.

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