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\(E\)-theory is a special case of \(KK\)-theory. (English) Zbl 1045.19001

In an appropriate realization Kasparov’s KK-theory group \(KK(A,B)\) consists of all the homotopy classes of homomorphisms \(KK(A,B) = [SA, \mathcal Q(B) \otimes \mathcal K]\), where \(SA := C_0(0,1) \otimes A\) is the suspension of the C*-algebra \(A\) and \(\mathcal Q(B) := M(B)/B\) is the Calkin algebra as the quotient of the double multiplier algebra \(M(B)\) by the C*-algebra \(B\) as an ideal and \(\mathcal K\) is the elementary C*-algebra of all the compact operators in a fixed separable Hilbert space. The theory is rather good as a K-bifunctor, but is not half-exact in general G. Skandalis [C. R. Acad. Sci., Paris, Sér. I, Math. 313, No.13, 939–941(1991; Zbl 0744.46066)].
A. Connes and N. Higson have introduced another variant of the KK-theory, the so called E-theory, which is half-exact with respect to arbitrary extensions of separable C*-algebras: by definition \(E(A,B) = [[A, \mathcal Q(B)\otimes \mathcal K]]\) consists of all the homotopy classes of asymptotic homomorphisms.
The relation between the two variants of the KK-theory is well-known as a natural map \[ KK(A,B) \to E(A,B), \] which is an isomorphism when \(A\) is nuclear, but which is not always injective because of half-exactness of the second theory and the non-half-exactness of the first. Applying this relation to the particular case of the Corona algebra \(\mathcal Q(B\otimes \mathcal K)= M(B\otimes \mathcal K)/B\otimes \mathcal K\) one has \[ KK(SA, \mathcal Q(B\otimes \mathcal K)) \to E(SA,\mathcal Q(B\otimes \mathcal K). \] “Thanks to the unrestricted excision properties of E-theory, combined with Bott-periodicity and stability, there is a natural isomorphism \[ E(SA, \mathcal Q(B\otimes \mathcal K)) \cong E(A,B), \] and we have therefore a natural map \[ KK(SA,\mathcal Q(B\otimes \mathcal K)) \to E(A,B). \]
With a rather long and rather complicated technique the author proves in the paper under review that this homomorphism is indeed an isomorphism in the case where the C*-algebra \(A\) is separable and the C*-algebra \(B\) is \(\sigma\)-unital (Theorem 5.8). The main task of the authors is to construct a group homomorphism \[ [[SA,\mathcal Q(B)\otimes \mathcal K]]\to [SA,\mathcal Q(B\otimes\mathcal K)\otimes\mathcal K].\text{''} \]

MSC:

19K35 Kasparov theory (\(KK\)-theory)
19K33 Ext and \(K\)-homology
46M15 Categories, functors in functional analysis

Citations:

Zbl 0744.46066
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