Geodesic
Translation Invariant Asymptotic Homomorphisms and Extensions of $C^*$-Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 3, pp. 87-91

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Let $A$ and $B$ be $C^*$-algebras, let $A$ be separable, and let $B$ be $\sigma$-unital and stable. We introduce the 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 $B$ from 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.
Keywords: $C^*$-algebra, asymptotic homomorphism, extension of $C^*$-algebras, homotopy equivalence of extensions, homotopy equivalence of asymptotic homomorphisms.
Mots-clés : Connes–Higson construction 
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     author = {V. M. Manuilov and K. Thomsen},
     title = {Translation {Invariant} {Asymptotic} {Homomorphisms} and {Extensions} of $C^*${-Algebras}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     publisher = {mathdoc},
     volume = {39},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_3_a9/}
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V. M. Manuilov; K. Thomsen. Translation Invariant Asymptotic Homomorphisms and Extensions of $C^*$-Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 3, pp. 87-91. http://geodesic.mathdoc.fr/item/FAA_2005_39_3_a9/