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The Baum-Connes conjecture. (English) Zbl 0911.46041
Let \(G\) be a group which is locally compact, Hausdorff and second countable. The group \(C^*\)-algebra, denoted \(C^*(G)\), is the enveloping \(C^*\)-algebra of the convolution algebra \(L_1(G)\). The reduced \(C^*\)-algebra of \(G\), denoted \(C^*_{\text{red}}(G)\), is the completion of \(L_1(G)\) in its regular representation as bounded operators on \(L_2(G)\). (The \(C^*\)-algebra \(C^*_{\text{red}}(G)\) coincides with \(C^*(G)\) iff \(G\) is amenable.) The purpose of this report is to present a short account of past and recent work on the Baum-Connes conjecture concerning the groups \(K_*(C^*(G))\) and \(K_*(C^*_{\text{red}}(G))\). The Baum-Connes conjecture for torsion-free groups is the following: if \(G\) is a discrete and torsion-free group then the assembly map \(\mu_{\text{red}}: K_*(BG)\to K_*(C^*_{\text{red}}(G))\), obtained from the so-named assembly map \(\mu: K_*(BG)\to K_*(C^*(G))\) using the regular representation \(C^*(G)\to C^*_{\text{red}}(G)\), is an isomorphism. The advantage of the new version of this conjecture is that it is simpler and applies more generally than the earlier statement. Associated to any \(G\) there is a proper \(G\)-space \({\mathcal E}G\), which is universal in the sense that any other proper \(G\)-space maps into it in a way which is unique up to equivariant homotopy. Using Kasparov’s KK-theory Baum, Connes and Higson formed the equivariant \(K\)-homology group \(K^G_*({\mathcal E}G)\). A class in \(K_*({\mathcal E}G)\) is represented by an abstract \(G\)-equivariant elliptic operator on \({\mathcal E}G\). If \(G\) is discrete and torsion-free then \({\mathcal E}G\) is universal principal space \(EG\) and \(K^G_*({\mathcal E}G)= K_*(BG)\). For general \(G\) there is an assembly map \(m_{\text{red}}: K^G_*({\mathcal E}G)\to K_*(C^*_{\text{red}}(G))\), very similar to the one already considered: a cycle for \(K^G_*({\mathcal E}G)\) is abstract elliptic operator \(D\) on a proper \(G\)-space \({\mathcal E}G\) and \(m_{\text{red}}\) associates to \(D\) its equivariant index.
The new Baum-Connes-Higson conjecture is the following: if \(G\) is any second countably, locally compact group then \(m_{\text{red}}\) is an isomorphism.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K56 Index theory
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