Bivariant \(K\)-theory and the Novikov conjecture
type: Item
instance of: scholarly article
title: Bivariant \(K\)-theory and the Novikov conjecture
zbMATH Open document ID: 0962.46052
DOI: 10.1007/PL00001630
published in: Geometric and Functional Analysis. GAFA
publication date: 2001-05-30T00:00:00Z
review text: It is well known that the injectivity of the Baum-Connes assembly map implies the Novikov conjecture. In the paper under review the author used some results of Yu and of some of his previous results to show that if there is a topological amenable action of a countable discrete group \(G\) on some compact Hausdorff space, then for every separable \(G\)-C*-algebra \(A\), the Baum-Connes assembly map \[ \mu: KK^G(\mathcal {EG}, A) \to KK(\mathbb C, C^*_r(G,A)) \] is split injective (Theorem 1.1). The author also proved (Theorem 1.2) a version of the Carlson-Pedersen theorem adapted to the Baum-Connes assembly map: For a countable discrete group \(G\) if the classifying space \(\mathcal {EG}\) is \(G\)-compact and admits a metrizable compactification \(\overline{\mathcal {EG}}\) which is a \(G\)-space and \(H\)-equivariantly contractible for every finite subgroup \(H\) of \(G\) and if \(\lim_{g \to \infty} \text{diam}_{\overline{\mathcal{EG}}} (gK) = 0\) for every compact subset \(K\) of \(\mathcal {EG}\) then the Baum-Connes assembly map is injective.
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 1525495
zbMATH Keywords: KK-group
zbMATH Keywords: assembly map
zbMATH Keywords: classification space
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: higher signature
zbMATH Keywords: Carlson-Pedersen theorm
Wikidata QID: Q123222374
author: Nigel Higson
MaRDI profile type: Publication
full work available at URL: pl00001630
OpenAlex ID: W1974598548
description: scientific article; zbMATH DE number 1525495