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

Details The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent

2007-07-26

arxiv.org/abs/0707.3861v2

J. Operator Theory, Volume 63, Issue 1 (2010), pp. 85-100

Mathematics

Operator Algebras

12 pages

Scientific paper

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 exist asymptotic morphisms from each to the other whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from KK-theory to E-theory. One more corollary is the following. T. Loring ([3]) proved that 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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