The coarse Baum-Connes conjecture and groupoids. II
type: Item
instance of: scholarly article
title: The coarse Baum-Connes conjecture and groupoids. II
zbMATH Open document ID: 1268.19001
published in: The New York Journal of Mathematics
publication date: 2012-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.0407
full work available at URL: 18-1.html
reviewed by: Liliana Răileanu
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 6032698
zbMATH Keywords: coarse geometry
zbMATH Keywords: groupoid
zbMATH Keywords: Baum-Connes conjecture
author: Jean-Louis Tu
MaRDI profile type: Publication
arXiv ID: 1005.0407
Recommended article: The coarse Baum-Connes conjecture and groupoids
Recommended article: Q4762254
Recommended article: The coarse Baum-Connes conjecture for relatively hyperbolic groups
Recommended article: Q4956097
Recommended article: Baum-Connes conjecture and coarse geometry
Recommended article: Coarse groupoid cohomology and Connes-Chern character
Recommended article: Coarse Baum-Connes conjecture
Recommended article: The Baum-Connes conjecture with coefficients for hyperbolic groups
Recommended article: The coarse Baum-Connes conjecture via \(C_{0}\) coarse geometry
Recommended article: Spaces of graphs, boundary groupoids and the coarse Baum-Connes conjecture
review text: Let \((X,d)\) be a metric space, proper (i.e., all closed balls are compact), not necessarily discrete. A subset \(E\) of \(X\times X\) is controlled if \(d_{|E}\) is bounded. In a previous paper by \textit{G. Skandalis} et al., see Part I [Topology 41, No. 4, 807--834 (2002; Zbl 1033.19003)], it was shown that \(G(X)=\bigcup_{E\text{\;controlled}}\) \(\overline{E}\subset \beta(X\times X)\) can be endowed with the structure of an étale, locally compact, \(\sigma-\)compact groupoid. Moreover, the coarse Baum-Connes conjecture for \(X\), see \textit{G. Yu} [K-Theory 9, No. 3, 199--221 (1995; Zbl 0829.19004)], which states that a certain assembly map NEWLINE\[NEWLINE\lim_d K_*(P_d(X))\to K(C^*(X))NEWLINE\]NEWLINE is an isomorphism, is equivalent to the Baum-Connes conjecture for \(G(X)\) with coefficients in \(\ell^\infty(X,{\mathcal K})\).NEWLINENEWLINEIn this paper, the main result from that paper is extended in the following two directions: first, the construction is extended to a large class of locally compact, proper metric spaces that are not necessarily discrete; secondly, a coarse Baum-Connes with coefficients is defined. The author shows that it is stable under taking closed subspaces.
description: scientific article; zbMATH DE number 6032698