Operator norm localization property of metric spaces under finite decomposition complexity
type: Item
instance of: scholarly article
title: Operator norm localization property of metric spaces under finite decomposition complexity
zbMATH Open document ID: 1185.46011
author: Xiaoman Chen
author: Qin Wang
author: Xianjin Wang
published in: Journal of Functional Analysis
publication date: 2009-10-20T00:00:00Z
review text: The notions of operator norm localization property and finite decomposition complexity were introduced to study the coarse Novikov conjecture and the stable Borel conjecture. The notion of a finite decomposition complexity was inspired by Gromov's definition of asymptotic dimension (see page 29 in [\textit{M.\,Gromov}, ``Geometric group theory.\ Volume 2: Asymptotic invariants of infinite groups'' (London Mathematical Society Lecture Note Series 182; Cambridge University Press) (1993; Zbl 0841.20039)]): a metric space \(X\) has finite asymptotic dimension if there is an integer \(n\geq 0\) such that, for any \(r >0\), the space \(X\) can be represented as a union of \(n + 1\) subsets \(X_i\), each of which may be further decomposed \(X_i=\bigcup_{i=1}^\infty X_{ij}\) with \(\text{dist}(X_{ij},X_{ik})> r\), and the sets \(\{X_{ij}\}_{i,j}\) have uniformly bounded diameters. This definition was generalized by \textit{E.\,Guentner}, \textit{R.\,Tessera} and \textit{G.-L.\thinspace Yu} [``Decomposition complexity and topological rigidity'' (Preprint) (2009)] in the following way: A family \(\mathcal{X}\) of metric spaces is \((n,r)\)-decomposable over a class \(\mathcal{U}\) of families \(\mathcal{Y}\) of metric spaces if every \(X\in \mathcal{X}\) admits a decomposition of the form described above with the requirement of uniformly bounded diameters replaced by the existence of \(\mathcal{Y}\in\mathcal{U}\) such that \(X_{ij}\in\mathcal{Y}\). \textit{J.\,Roe} [``Coarse cohomology and index theory on complete Riemannian manifolds'', Mem.\ Am.\ Math.\ Soc.\ 497 (1993; Zbl 0780.58043)] introduced the following notion: Let \(X\) be a metric space with a positive locally finite Borel measure \(\nu\), and let \(H\) be a separable Hilbert space. A bounded linear operator \(T: L_2(X,\nu)\otimes H\to L_2(X,\nu)\otimes H\) is said to have propagation at most \(r\) if, for all \(\varphi,\psi\in L_2(X,\nu)\otimes H\) with bounded supports such that \(\text{dist}(\text{supp}\,\varph…
Mathematics Subject Classification ID: 46B85
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 5618761
zbMATH Keywords: coarse Novikov conjecture
zbMATH Keywords: finite decomposition complexity
zbMATH Keywords: linear group
zbMATH Keywords: metric space
zbMATH Keywords: operator norm localization
reviewed by: Mikhail Ostrovskii
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.05.022
OpenAlex ID: W1974169633
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DOI: 10.1016/J.JFA.2009.05.022
description: scientific article; zbMATH DE number 5618761