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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