\(E\)-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.

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.

Reviewer: V.M.Deundjak (Rostov-na-Donu)