Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms - Manuilov, V.; Thomsen, K. (Mathematica Scandinavica, 2007)
Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $σ$-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.