Operator norm localization property of relative hyperbolic group and graph of groups
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 Note that Higson, in an unpublished paper, using some kind of localization estimation of operator norm, gave a counterexample to the coarse Baum-Connes conjecture.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 \(X\) has bounded geometry and for all \(t>0\) there exists \(Y_{t}\subset X\) such that \(\{\Gamma_{\alpha}\backslash Y_{t}\}\) is \(t-\)disjoint and \(Y_{t}\) has operator norm localization property with constant \(c.\) Then \(X\) has the operator norm localization property with constant \((1+\varepsilon)c\) for any \(\varepsilon>0.\) Finally, the authors give two interesting applications of the main Theorem. First, they prove that a finitely generated group \(\Gamma\) which is strongly hyperbolic with respect to a collection of finitely generated subgroups \(\{H_{1},\dots,H_{n}\}\) has the operator norm localization property if and only if each \(H_{i}\) has the operator norm localization property. Furthermore, they prove that, if \(\pi\) is the fundamental group of a connected finite graph of groups with finitely generated vertex groups \(G_{P} \) and if \(G_{P}\) has the operator norm localization property for all vertices \(P, \) then \(\pi\) has the operator norm localization property.
scientific article; zbMATH DE number 5318662
scientific article