Extensions of C*-algebras and translation invariant asymptotic homomorphisms

Details Extensions of C*-algebras and translation invariant asymptotic homomorphisms Extensions of C*-algebras and translation invariant asymptotic homomorphisms

2003-10-31

arxiv.org/abs/math/0310492v1

Mathematics

Operator Algebras

22 pages, LaTeX, XYpic

Scientific paper

Let $A$, $B$ be C*-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathbb R)\otimes A$ 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.

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