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$$E$$-theory and $$KK$$-theory for groups which act properly and isometrically on Hilbert space. (English) Zbl 0988.19003
Invent. Math. 144, No. 1, 23-74 (2001); correction J. Noncommut. Geom. 13, No. 2, 797-803 (2019).
Let $$G$$ be a group which is locally compact, Hausdorff and second countable. The full group $$C^*$$-algebra, denoted $$C^*_{\max}(G),$$ 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 $$C^*_{\max}(G)$$ iff $$G$$ is amenable.) 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 $$\mu_{\max}: K_*(BG)\to K_*(C^*_{\max}(G)),$$ is an isomorphism.
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 $$K_*^G({\mathcal E}G).$$ If $$G$$ is discrete and torsion-free then $${\mathcal E}G$$ is the universal principal space $$EG$$ and $$K_*^G({\mathcal E}G)=K_*(BG).$$ For general $$G$$ there are assembly maps $$\mu_{\text{red}}: K_*^G({\mathcal E}G)\to K_*(C^*_{\text{red}}(G)),$$ $$\mu_{\max}: K_*^G({\mathcal E}G)\to K_*(C^*_{\max}(G))$$ (a cycle for $$K_*^G({\mathcal E}G)$$ is an abstract elliptic operator $$D$$ on a proper $$G$$-space $${\mathcal E}G$$ and $$\mu_{\text{red}}$$ associates to $$D$$ its equivariant index). The new Baum-Connes-Higson conjecture is the following: if $$G$$ is any second countable, locally compact group then $$\mu_{\text{red}}$$ is an isomorphism.
Both the assembly maps $$\mu_{\text{red}}$$ and $$\mu_{\text{max}}$$ may be generalized by introducing a coefficient $$C^*$$-algebra $$A,$$ on which $$G$$ acts continuously by $$C^*$$-algebra automorphisms. If the associated reduced crossed product $$C^*$$-algebra is denoted $$C^*_{\text{red}}(G,A)$$ then there is a reduced assembly map $$\mu_{\text{red}}: K_*^G({\mathcal E}G,A)\to K_*(C^*_{\text{red}}(G,A))$$ defined using KK-theory, along with a similar map $$\mu_{\max}$$ for the full crossed product algebra $$C^*_{\text{red}}(G,A).$$ The main purpose of this article is to prove the Baum-Connes conjecture for an interesting and fairly broad class of groups which are known to harmonic analysts as groups with the Haagerup approximation property. THEOREM. If $$G$$ is a second countable, locally compact group, and if $$G$$ has the Haagerup approximation property, then for any separable $$G-C^*$$-algebra $$A$$ the Baum-Connes assembly maps $$\mu_{\text{red}}: K_*^G({\mathcal E}G,A)\to K_*(C^*_{\text{red}}(G,A)),$$ $$\mu_{\max}: K_*^G({\mathcal E}G,A)\to K_*(C^*_{\max}(G,A))$$ are isomorphisms of abelian groups.

##### MSC:
 19K56 Index theory 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K35 Kasparov theory ($$KK$$-theory) 58B34 Noncommutative geometry (à la Connes)
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