The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space
type: Item
instance of: scholarly article
title: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space
zbMATH Open document ID: 0956.19004
published in: Inventiones Mathematicae
publication date: 2001-03-07T00:00:00Z
review text: First of all let us recall the coarse Baum-Connes conjecture. A metric space is called proper if every closed ball \(B(x,r)\) is compact. Let \((M,d)\) be a proper metric space and \(H_M\) a separable Hilbert space equipped with a faithful and non-degenerate *-representation of \(C_0(M)\) whose range contains no non-zero compact operator. A bounded operator \(T:H_M\to H_M\) has finite propagation if \(\sup \{d(m_1,m_2):(m_1,m_2)\in \text{supp}(T)\} <\infty \) where \(m_i \in M\) and \(\text{supp}(T)\) is the complement of the set of points \((m_1,m_2)\in M\times M\) for which there exist \(g_1,g_2 \in C_0(M)\) such that \(g_1Tg_2=0\), \(g_1(m_1)\neq 0\), \(g_2(m_2)\neq 0.\) A bounded operator \(T:H_M\to H_M\) is called locally compact if the operators \(gT\) and \(Tg\) are compact for all \(g\in C_0(M).\) The Roe algebra \(C^*(M)\) is the operator norm closure of the *-algebra of all locally compact, finite propagation operators acting on \(H_M.\) A metric space is called locally finite if every ball contains finitely many elements. Let \(\Gamma\) be a locally finite discrete metric space. For each \(d\geq 0,\) the Rips complex \(P_d(\Gamma)\) is the simplicial polyhedron where the set of all vertices is \(\Gamma ,\) and a finite subset \(\{\gamma _0,\dots,\gamma _n\}\subset \Gamma\) spans a simplex iff \(d(\gamma _i ,\gamma _j) \leq d\) for all \(0\leq i,j \leq n.\) Endow \(P_d(\Gamma)\) with the spherical metric. Recall that a discrete metric space \(X\) is said to have bounded geometry if for all \(r>0\), there exists \(N(r)>0 \) such that the number of elements in \(B(x,r)\) is at most \(N(r)\) for all \(x\in X.\) Every finitely generated group, as a metric space with a word-length metric, has bounded geometry. The coarse Baum-Connes conjecture: if \(\Gamma\) is a discrete metric space with bounded geometry, then the index map from \(\lim _{d\to \infty} K_*(P_d(\Gamma))\) to \(\lim _{d\to \infty} K_*(C^*(P_d(\Gamma)))\) is an isomorphism, where \(K_*(P_d(\Gamma)…
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 1388911
zbMATH Keywords: operator K-theory
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: index theory
zbMATH Keywords: KK-theory
zbMATH Keywords: Roe algebra
zbMATH Keywords: Rips complex
zbMATH Keywords: spherical metric
zbMATH Keywords: bounded geometry
Wikidata QID: Q122919879
author: Guoliang Yu
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: s002229900032
OpenAlex ID: W2024527795
DOI: 10.1007/S002229900032
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description: scientific article; zbMATH DE number 1388911