Counterexamples to the Baum-Connes conjecture
type: Item
instance of: scholarly article
title: Counterexamples to the Baum-Connes conjecture
zbMATH Open document ID: 1014.46043
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-07-29T00:00:00Z
review text: To an action of a group \(G\) on a \(C^*\)-algebra \(A\) one can associate topological \(K\)-theory groups \(K^{\text{top}}_*(G;A)\) and operator \(K\)-theory groups \(K_*(A\rtimes_r G)\), which are related by the assembly map \(\mu_A:K^{\text{top}}_*(G;A)\to K_*(A\rtimes_r G)\). The Baum-Connes conjecture with coefficients asserts that \(\mu_A\) is an isomorphism. It is a generalization of the classical Baum-Connes conjecture, which asserts the same for \(A={\mathbb C}\). Another generalization of the conjecture uses groupoids instead of groups. The authors provide counterexamples to: injectivity and surjectivity of the assembly map for Hausdorff groupoids; injectivity and surjectivity of the assembly map for holonomy groupoids of foliations; surjectivity of the assembly map for coarse geometric spaces; surjectivity of the assembly map for discrete group actions on commutative \(C^*\)-algebras. The main idea is that the failure of exactness, when short exact sequences of algebras are completed to reduced \(C^*\)-algebras, can be detected at the level of \(K\)-theory, where the Baum-Connes conjecture predicts exactness.
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 1773367
zbMATH Keywords: group \(C^*\)-algebra
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: assembly map
author: Nigel Higson
author: Vincent Lafforgue
author: Georges Skandalis
reviewed by: Vladimir M. Manuilov
MaRDI profile type: Publication
Wikidata QID: Q59701765
full work available at URL: s00039-002-8249-5
OpenAlex ID: W2135995235
DOI: 10.1007/S00039-002-8249-5
description: scientific article; zbMATH DE number 1773367