\(K\)-theory for Sp\((n,1)\)
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.
scientific article; zbMATH DE number 2160160
scientific article