Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms

Authors

  • V. Manuilov
  • K. Thomsen

DOI:

https://doi.org/10.7146/math.scand.a-15018

Abstract

Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathsf{R})\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.

Downloads

Published

2007-03-01

How to Cite

Manuilov, V., & Thomsen, K. (2007). Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms. MATHEMATICA SCANDINAVICA, 100(1), 131–160. https://doi.org/10.7146/math.scand.a-15018

Issue

Section

Articles