The Connes–Higson construction is an isomorphism

Let A be a separable C∗-algebra and B a stable C∗-algebra containing a strictly positive element. We show that the group Ext−1/2(SA,B) of unitary equivalence classes of extensions of SA by B, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes–Higson construction gives rise to an isomorphism between Ext−1/2(SA,B) and the E-theory group E(A,B) of homotopy classes of asymptotic homomorphisms from S2A to B.