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(R)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.