Operator norm localization property of relative hyperbolic group and graph of groups
type: Item
instance of: scholarly article
title: Operator norm localization property of relative hyperbolic group and graph of groups
zbMATH Open document ID: 1152.53031
author: Xiaoman Chen
author: Xianjin Wang
published in: Journal of Functional Analysis
publication date: 2008-09-01T00:00:00Z
review text: This paper focuses on an operator norm localization property which generalizes the local estimation property in the case of finite dimensional metric spaces. Note that Higson, in an unpublished paper, using some kind of localization estimation of operator norm, gave a counterexample to the coarse Baum-Connes conjecture. Let \(X\) be a discrete metric space and \(H\) be a separable and infinite dimensional Hilbert space. A bounded operator \(T:l^{2}(X)\otimes H\rightarrow l^{2}(X)\otimes H,\) is said to have propagation at most \(r\) if for all \(\varphi,\psi\in l^{2}(X)\otimes H\) with \(d(\text{Supp}(\varphi), \text{Supp}(\psi))>r\) we have \(\langle T\varphi,\psi\rangle=0.\) The space of operators acting on \(l^{2}(X)\otimes H\) with propagation at most \(r\) will be denoted by \(\mathcal{A}_{r}(X).\) Let now \(f:\mathbb{N}\rightarrow\mathbb{N}\) be a non-decreasing function. We say that \(X\) has operator norm localization property relative to \(f\) with constant \(c\geq1\) if, for all \(k\in\mathbb{N},\) and every \(T\in\mathcal{A}_{r}(X),\) there exists nonzero \(\varphi\in l^{2}(X)\otimes H\) satisfying, \(\text{Diam(Supp}(\varphi))\leq f(k)\) and \(\|T\|\|\varphi\|\leq c\|T\varphi\|.\) A family of metric spaces \(\{\Gamma _{\alpha}\}_{\alpha\in J}\) is said to have uniform operator norm localization property if there exists a common constant \(c\geq1\) and a common function \(f:\mathbb{N}\rightarrow\mathbb{N}\) such that, for each \(\alpha\in J,\) \(\Gamma_{\alpha}\) has operator norm localization property relative to \(f\) with constant \(c.\) A discrete metric space \(X\) has bounded geometry, if for every \(R>0,\) there is a uniform bound on the number of elements in the ball of radius \(R\) in \(X.\) The main theorem proved by the authors is the following. Let \(\{\Gamma _{\alpha}\}_{\alpha\in J}\) be a family of metric spaces which has the uniform operator norm localization property with constant \(c.\) Let \(X=\cup _{_{\alpha\in J}}\Gamma_{\alpha}.\) If…
reviewed by: Charalampos Charitos
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 19K56
zbMATH DE Number: 5318662
zbMATH Keywords: operator norm localization property
zbMATH Keywords: coarse invariant
zbMATH Keywords: Roe algebras
zbMATH Keywords: finite propagation
zbMATH Keywords: strongly relative hyperbolic group
zbMATH Keywords: graph of groups
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.04.022
OpenAlex ID: W1996647207
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DOI: 10.1016/J.JFA.2008.04.022
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description: scientific article; zbMATH DE number 5318662