\(K\)-theory for Sp\((n,1)\)
type: Item
instance of: scholarly article
title: \(K\)-theory for Sp\((n,1)\)
zbMATH Open document ID: 1082.46050
published in: Journal of Functional Analysis
publication date: 2005-04-21T00:00:00Z
review text: For a locally compact group \(G\), let \(\lambda:C^*(G)\to C^*_r(G)\) be the canonical map from the full group \(C^*\)-algebra onto the reduced one defined by the regular representation of \(G\). The Baum-Connes conjecture predicts the \(K\)-theory group of \(C^*_r(G)\) but not that of \(C^*(G)\) and the map \(\lambda_*:K_*(C^*(G))\to C^*_r(G)\) may not be an isomorphism. The author shows how far \(\lambda_*\) is from an isomorphism for \(G=Sp(n,1)\). Let \(I\) denote the set of ``isolated'' series representations of \(G\). The trivial representation is one of them; it is the only one when \(n=2\) and there are countably many others, which are infinite dimensional, when \(n>2\). Let \(p=\lambda\oplus(\oplus_I\pi)\) be the direct sum of the regular representation and of all representations from the ``isolated'' series. It is proved that \(p:C^*(G)\to C^*_r(G)\oplus(\bigoplus_{\pi\in I}{\mathcal K}(H_\pi))\) induces a \(KK\)-equivalence. The author also computes the range of the full Baum-Connes map \(R(K)\to K_0(C^*(G))\), where \(K\) is a maximal compact subgroup of \(G\) and \(R(K)\) denotes its representation ring.
reviewed by: Vladimir M. Manuilov
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 2160160
zbMATH Keywords: group \(C^*\)-algebra
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Baum-Connes map
author: Nicolas Prudhon
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.05.011
OpenAlex ID: W1993685142
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DOI: 10.1016/J.JFA.2004.05.011
description: scientific article; zbMATH DE number 2160160