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extensions of c*-algebras

Proceedings of the London Mathematical Society (1)

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$E$-Theory is a Special Case of $KK$-Theory
V. Manuilov, K. Thomsen
Journal:

Proceedings of the London Mathematical Society / Volume 88 / Issue 2 / March 2004

Published online by Cambridge University Press:

08 March 2004, pp. 455-478

Print publication:

March 2004
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Let $A$ and $B$ be C*-algebras, with $A$ separable and $B$$\sigma$-unital and stable. It is shown that there are natural isomorphisms
$$ E(A,B) = KK(SA,Q(B)) = [SA,Q(B) \otimes K], $$
where $SA=C_0(0, 1) \otimes A$, $[\cdot, \cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B) / B$ is the generalized Calkin algebra, and $K$ denotes the C*-algebra of compact operators of an infinite-dimensional separable Hilbert space.