Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms
Keyword(s):
Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathsf{R})\otimes A$ 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.
1989 ◽
Vol 41
(6)
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pp. 1021-1089
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1998 ◽
Vol 111
(1)
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pp. 7-37
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2005 ◽
Vol 39
(3)
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pp. 236-239
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2016 ◽
Vol 27
(03)
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pp. 1650023
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2019 ◽
Vol 38
(5)
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pp. 215-232
Keyword(s):
2020 ◽
Vol 35
(22)
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pp. 2050118
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2013 ◽
Vol 11
(04)
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pp. 1360003
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