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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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