Uniform \(K\)-homology theory
type: Item
instance of: scholarly article
title: Uniform \(K\)-homology theory
zbMATH Open document ID: 1173.46049
author: Ján Špakula
published in: Journal of Functional Analysis
publication date: 2009-06-30T00:00:00Z
review text: The author modifies the construction of analytic \(K\)-homology for metric spaces by introducing uniformity of approximation for \(K\)-homology cycles. A bounded operator \(T\) on a Hilbert space \(H\) is said to be \((\varepsilon,M)\)-approximable if there is a rank \(M\) operator \(k\) such that \(\|T-k\|<\varepsilon\). Let \(X\) be a metric space, \(C_R(X)=\{f\in C_c(X):\text{diam}(\text{supp}(f))\leq R; \|f\|\leq 1\}\), and let \(E:C_R(X)\to\mathbb B(H)\) be a linear map. \(E\) is \((\varepsilon,R,M)\)-approximable if \(E(f)\) is \((\varepsilon,M)\)-approximable for any \(f\in C_R(X)\). \(E\) is uniformly approximable if for every \(R\geq 0\), \(\varepsilon>0\) there exists \(M>0\) such that \(E\) is \((\varepsilon,R,M)\)-approximable. Let \(\varphi:C_0(X)\to\mathbb B(H)\) be a \(*\)-homomorphism. A triple \((H,\varphi,T)\) defines a \(K\)-homology cycle if several linear maps from \(C_0(X)\) to \(\mathbb B(H)\) (like \(f\mapsto [T,\varphi(f)]\), \((1-T^*T)\varphi(f)\), etc.), take values in the space of compact operators. The author defines uniform cycles by requiring that these linear maps should also be uniformly approximable. These uniform cycles are used to define uniform analytic \(K\)-homology theory in the standard way. Then the author defines an index map from this theory to the \(K\)-theory of uniform Roe \(C^*\)-algebras analogous to the coarse assembly map, shows that, for a torsion-free countable group \(\Gamma\), the direct limit \(\lim_{d\to\infty}K^u_*(P_d\Gamma)\) of the uniform \(K\)-homology of the Rips complexes \(P_d(\Gamma)\) of \(\Gamma\) is isomorphic to \(K^{\text{top}}_*(\Gamma,l^\infty\Gamma)\), and proves a criterion for amenability in terms of vanishing of a `fundamental class' similar to that in the uniformly finite homology.
reviewed by: Vladimir M. Manuilov
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K33
zbMATH DE Number: 5572188
zbMATH Keywords: analytic \(K\)-homology
zbMATH Keywords: coarse assembly map
zbMATH Keywords: uniform Roe algebra
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.02.008
OpenAlex ID: W2011756913
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DOI: 10.1016/J.JFA.2009.02.008
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description: scientific article; zbMATH DE number 5572188