Asymptotic homomorphisms and equivariant \(KK\)-theory
The relation between \(KK\)-theory of Kasparov and \(E\)-theory \[ E(A,B)= [[SA\otimes{\mathcal K}, SB\otimes{\mathcal K}]] \] [cf. \textit{A. Connes} and \textit{N. Higson}, C. R. Acad. Sci., Paris, Sér. I 311, No. 2, 101-106 (1990; Zbl 0717.46062)] based on asymptotic homomorphisms \(\varphi= (\varphi_t)_{t\in [1,\infty)}: SA\otimes{\mathcal K}\to SB\otimes{\mathcal K}\), where the individual maps, the \(\varphi_t\)'s, can be completely arbitrary, but satisfy the algebraic identities defining a \(*\)-homomorphism asymptotically as \(t\) tends to infinity, was clarified by \textit{T. Houghton-Larsen} and the author in a recent paper [``Universal (co) homology theories'', \(K\)-theory 16, No. 1, 1-27 (1999)]. Namely, it was proved that: \[ KK(A, B)= [[SA\otimes{\mathcal K}, SB\otimes{\mathcal K}]]_{cp}, \] where the right-hand side differs from \(E(A,B)\) in that the only asymptotic homomorphisms considered are those for which the individual maps, \(\varphi_t\)'s are all completely positive linear contractions. In the present paper, the author generalizes this result to cover the equivalent \(KK\)-theory of \textit{G. G. Kasparov} [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. The main result proves that for any locally compact second countable group \(G\), there is an isomorphism \[ KK^G(A, B)\simeq [[SA\otimes\widetilde{\mathcal K}_G, SB\otimes\widetilde{\mathcal K}_G]]^G_{cp}, \] where \(\widetilde{\mathcal K}_G\) is the \(G\)-algebra of compact operators obtained by considering the direct sum of countably many copies of the regular representation of \(G\), and the asymptotic homomorphisms consist of those for which the individual maps are completely positive linear and equivariant contractions. The isomorphism is natural in both variables and takes the Kasparov product to the obvious defined composition product.
scientific article; zbMATH DE number 1286371
scientific article