Journal of Operator Theory
Volume 63, Issue 1, Winter 2010 pp. 85-100.
The C∗-algebras qA⊗K and S2A⊗K are asymptotically equivalentAuthors: 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 S2A⊗K and the C∗-algebra qA⊗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 S2A⊗K to qA⊗K and an asymptotic morphism from qA⊗K to S2A⊗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 ∗-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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