Operator norm localization property of metric spaces under finite decomposition complexity
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 \(\text{dist}(\text{supp}\,\varphi, \text{supp}\,\psi) > r\), we have \(\langle T\varphi, \psi\rangle = 0\). \textit{X.-M.\thinspace Chen}, \textit{R.\,Tessera}, \textit{X.-J.\thinspace Wang} and \textit{G.-L.\thinspace Yu} [``Metric sparsification and operator norm localization'', Adv.\ Math.\ 218, No.\,5, 1496--1511 (2008; Zbl 1148.46040)] introduced the following notion: A family \(\mathcal{X}\) of metric spaces is said to have uniform operator norm localization property if there exist \(c\in (0, 1]\) and a nondecreasing function \(f: \mathbb{R}_+\to\mathbb{R}_+\) such that, for each \(X\in\mathcal{X}\), every positive locally finite Borel measure \(\nu\) on \(X\), every \(r>0\), and every \(T: L_2(X,\nu)\otimes H\to L_2(X,\nu)\otimes H\) with propagation at most \(r\), there exists a nonzero vector \(\xi\in L_2(X,\nu)\otimes H\) satisfying (1) \(|T\xi|\geq c|T|\,|\xi|\); (2) \(\text{diam}(\text{supp}\,\xi)\leq f(r)\). The collection of families of metric spaces having \textit{weak finite decomposition complexity with respect to the operator norm localization property} is defined to be the smallest collection containing families having uniform operator norm localization property and closed under the decomposability described above. The main result of the paper is as follows. A metric space \(X\) which has weak finite decomposition complexity with respect to the operator norm localization property has the operator norm localization property (the latter is defined as above for \(\mathcal{X}=\{X\}\)).
scientific article; zbMATH DE number 5618761