Asymptotically split extensions and \(E\)-theory (Q2761458)

The aim of this paper is related to the Connes-Higson construction \((E\)-theory) [\textit{A. Connes} and \textit{N. Higson}, C. R. Acad. Sci., Paris, Sér. I Math. 311, 101-106 (1990; Zbl 0717.46062)] which produces an asymptotic homomorphism from an extension of \(C^*\)-algebras and so provides a way for the study of \(C^*\)-extensions via asymptotic homomorphisms. Namely, it is shown that the \(E\)-theory admits a formulation in terms of \(C^*\)-extensions, in much the same way as the \(KK\)-theory of Kasparov. The essential new feature of this description of \(E\)-theory is that the role played by split extensions in the definition of the \(KK\)-theory is taken here by so-called asymptotically split extensions: All extension of a \(C^*\)-algebra \(A\) by a stable \(C^*\)-algebra \(B\) is said to be asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is semi-invertible if it can be made asymptotically split by adding another extension to it.NEWLINENEWLINENEWLINEThe main result of the paper says that there exists a one-to-one correspondence between the asymptotic homomorphisms from \(SA\) to \(B\) and the homotopy classes of semi-invertible extensions of \(S^2A\) by \(B\).