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# 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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