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Journal of Operator Theory

Volume 63, Issue 1, Winter 2010  pp. 85-100.

The $C^*$-algebras $qA\otimes \mathcal{K}$ and $S^2A\otimes \mathcal{K}$ are asymptotically equivalent

Authors Tatiana Shulman
Author institution: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark

Summary:  Let $A$ be a separable $C^*$-algebra. We prove that its stabilized second suspension $S^2A\otimes \mathcal K$ and the $C^*$-algebra $qA\otimes \mathcal K$ constructed by Cuntz in the framework of his picture of $KK$-theory are asymptotically equivalent. This means that there exists an asymptotic morphism from $S^2A\otimes \mathcal K$ to $qA\otimes \mathcal K$ and an asymptotic morphism from $qA\otimes \mathcal K$ to $S^2A\otimes \mathcal K$ whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from $KK$-theory to $E$-theory. Also by Loring's result any asymptotic morphism from $\qC$ to any $C^*$-algebra $B$ is homotopic to a $\ast$-homomorphism. We prove that the same is true when $\C$ is replaced by any nuclear $C^*$-algebra $A$ and when $B$ is stable.

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