Asymptotic homomorphisms and equivariant \(KK\)-theory
type: Item
instance of: scholarly article
title: Asymptotic homomorphisms and equivariant \(KK\)-theory
zbMATH Open document ID: 0929.46059
author: Klaus Thomsen
published in: Journal of Functional Analysis
publication date: 2000-01-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 1286371
zbMATH Keywords: \(KK\)-theory of Kasparov
zbMATH Keywords: \(E\)-theory
zbMATH Keywords: asymptotic homomorphisms
zbMATH Keywords: equivariant contractions
zbMATH Keywords: Kasparov product
zbMATH Keywords: composition product
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
full work available at URL: jfan.1998.3377
OpenAlex ID: W2008681965
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DOI: 10.1006/JFAN.1998.3377
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description: scientific article; zbMATH DE number 1286371