Comparison of \(KE\)-theory and \(KK\)-theory
type: Item
instance of: scholarly article
title: Comparison of \(KE\)-theory and \(KK\)-theory
zbMATH Open document ID: 1361.19002
DOI: 10.4171/JNCG/256
author: Ralf Meyer
published in: Journal of Noncommutative Geometry
publication date: 2016-11-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1503.04012
reviewed by: Bernhard Burgstaller
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 6655310
zbMATH Keywords: \(KK\)-theory
zbMATH Keywords: \(E\)-theory
zbMATH Keywords: \(KE\)-theory
MaRDI profile type: Publication
arXiv ID: 1503.04012
cites work: Q5748452
cites work: On an intermediate bivariant \(K\)-theory for \(C^*\)-algebras
cites work: Q3894484
cites work: Equivariant KK-theory and the Novikov conjecture
cites work: Equivariant Kasparov theory and generalized homomorphisms
OpenAlex ID: W3098531743
review text: In his Ph.D. Thesis and in [J. Noncommut. Geom. 10, No. 3, 1083--1130 (2016; Zbl 1353.46054)], \textit{C. D. Dumitraşcu} considers an equivariant bivariant \(K\)-theory called \(KE\)-theory which is by definition a mix of Kasparov's \(KK\)-theory and A. Connes and N. Higson's picture of \(E\)-theory. Indeed there are natural concrete maps NEWLINE\[NEWLINEKK^G(A,B) \rightarrow KE^G(A,B) \rightarrow E^G(A,B),NEWLINE\]NEWLINE where \(G\) is a second countable locally compact group.NEWLINENEWLINEThe author of this article was suspicious that \(KE\)-theory could be either isomorphic to \(KK\)-theory or \(E\)-theory, and shows that the map \(KE^G(A,B) \rightarrow E^G(A,B)\) factors through a map \(KE^G(A,B) \rightarrow KK^G(A,B)\), the map NEWLINE\[NEWLINEKK^G(A,B) \rightarrow KE^G(A,B) \rightarrow KK^G(A,B)NEWLINE\]NEWLINE is the identity map, and hence, \(KK^G\)-theory is a direct summand of \(KE^G\)-theory.NEWLINENEWLINEThe proof uses the equivariant Cuntz picture of \(KK^G\)-theory and is otherwise the usual technical work with cycles and completely positive equivariant asymptotic morphisms.
description: scientific article; zbMATH DE number 6655310