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

TL;DR

This paper demonstrates that the stabilized second suspension of a separable C*-algebra and a Cuntz-constructed algebra are asymptotically equivalent, simplifying the relationship between KK-theory and E-theory.

Contribution

It establishes the asymptotic equivalence between $S^2A\otimes \mathcal K$ and $qA\otimes \mathcal K$ for separable C*-algebras, extending known results to nuclear algebras.

Findings

Asymptotic morphisms exist in both directions between the algebras.

The composition of these morphisms is homotopic to the identity.

Simplifies the description of the natural transformation from KK-theory to E-theory.

Abstract

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.