Asymptotically split extensions and E-theory

Details Asymptotically split extensions and E-theory Asymptotically split extensions and E-theory

1999-11-26

arxiv.org/abs/math/9911208v1

Mathematics

Operator Algebras

14 pages, LaTeX

Scientific paper

We show that the E-theory of Connes and Higson can be formulated in terms of C*-extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is called semi-invertible if it can be made asymptotically split by adding another extension to it. Our main result is that there exists a one-to-one correspondence between asymptotic homomorphisms from $SA$ to $B$ and homotopy classes of semi-invertible extensions of $S^2A$ by $B$.

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