The Connes–Higson construction is an isomorphism

Abstract

Let A be a separable C Ã -algebra and B a stable C Ã -algebra containing a strictly positive element. We show that the group Ext À1=2 ðSA; BÞ 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 À1=2 ðSA; BÞ and the E-theory group EðA; BÞ of homotopy classes of asymptotic homomorphisms from S 2 A to B: r 2004 Elsevier Inc. All rights reserved.

Homotopy invariance in E-theory

“…Note that the last result improves on Theorem 3.4 of [20] in two ways: The asymptotic homomorphism ν is defined on cone(A), not only on SA, and the path of unitaries comes from M 2 (B)…”

“…[8]. Now a direct and quite simple proof exists and it was described in Theorem 3.4 of [20]. We can therefore concentrate here on the equivalence between (1) and (3).…”

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

“…This is possible by the arguments that prove the existence of quasi-central approximate units, [1]. {u n } ∞ n=0 is a unit sequence in the sense of [11] and [12]. We set ∆ 0 = √ u 0 and ∆ n = √ u n − u n−1 , n ≥ 1.…”

All extensions of $${C^*_r\left(\mathbb F_n\right)}$$ are semi-invertible

“…Motivated by this fact and the asymptotic homomorphism approach to extensions of Connes and Higson [3], Manuilov and the author introduced in [7] a theory of extensions which basically only differs from the conventional theory, introduced in the work of Brown et al [2], in that the split extensions are replaced by extensions which are only asymptotically split. It was shown in [7] that the resulting theory is a version of the E-theory of Connes and Higson when the algebra which plays the role of the quotient in the extensions is a suspended C * -algebra. Furthermore, it was shown in [7] that any extension of a suspended C * -algebra is semi-invertible in the sense that one can find another extension, namely the one obtained by reversing the orientation in the suspension, such that the addition of the two extensions is asymptotically split.…”