\(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory
type: Item
instance of: scholarly article
title: \(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory
zbMATH Open document ID: 0863.46045
author: Terry A. Loring
author: Marius Dadarlat
published in: Journal of Functional Analysis
publication date: 1997-06-03T00:00:00Z
review text: Let \(A\), \(B\) be separable \(C^*\)-algebras, \(SA\) be the suspension of \(A\), \(E(A,B)\) be Connes/Higson bivariant homology theory (the elements of \(E(A,B)=[[SA\otimes\text{Comp};SB\otimes\text{Comp}]]\) are taken to be homotopy classes of so named asymptotic monomorphisms from \(SA\otimes K\) to \(SB\otimes K\)). The main purpose of this paper is to prove that \(E(A,B)=[[A\otimes\text{Comp};B\otimes\text{Comp}]]\) for some \(A\) and all \(B\). The class of those \(A\) include the dimension-drop intervals \[ A_n=\{f\in C([0,1];L(n;\mathbb{C}))\mid f(0), f(1)\text{ are scalar}\}. \] In this case, there is no distinctions between homomorphisms and asymptotic morphisms, and so these results are used in the problem of classification of certain \(C^*\)-algebras. The dimension-drop interval may also be used to describe \(K_*\)-groups and \(KK\)-groups: 1) for \(B\) separable \(K_0(B;\mathbb{Z}/n)\cong[[A_n;B\otimes\text{Comp}]]\), 2) for a locally compact space \(X\) with \(X^+\) having the homotopy type of a finite CW-complex \(K_1(X;\mathbb{Z}/n)=K^1(C_0(X);\mathbb{Z}/n)\cong[[C_0(X),A_n\otimes\text{Comp}]]\), 3) for \(A\) and \(B\) separable and \(A\) nuclear \(KK(A,B;\mathbb{Z}/n)\cong[[A_n\otimes A,B\otimes\text{Comp}]]\).
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 763974
zbMATH Keywords: \(KK\)-theory
zbMATH Keywords: \(K\)-homology
zbMATH Keywords: separable \(C^*\)-algebras
zbMATH Keywords: suspension
zbMATH Keywords: Connes/Higson bivariant homology theory
zbMATH Keywords: homotopy classes
zbMATH Keywords: asymptotic monomorphisms
zbMATH Keywords: dimension-drop intervals
zbMATH Keywords: \(K_ *\)-groups
zbMATH Keywords: \(KK\)-groups
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: jfan.1994.1151
OpenAlex ID: W1979780052
DOI: 10.1006/JFAN.1994.1151
description: scientific article; zbMATH DE number 763974