A categorical perspective on the Atiyah-Segal completion theorem in \(KK\)-theory (Q1656218)

Many important results about the equivariant \(K\)-theory of spaces have analogues that concern the equivariant \(KK\)-theory of \(C^*\)-algebras with an action of a compact group. A first important example of this is the Universal Coefficient Theorem by Rosenberg and Schochet for \(G\)-equivariant the \(KK\)-theory, where \(G\) is a connected Lie group with torsion-free fundamental group. This article deals with such bivariant versions of the Atiyah--Segal completion theorem and McClure's restriction map theorem. The latter theorem is used to weaken the hypotheses that are needed to deduce that the Baum--Connes assembly map with coefficients is an isomorphism for an extension of groups. The method in this paper is homological algebra in triangulated categories, which is applied to the restriction functors from \(KK^G\) to \(KK^H\) for a family \({\mathcal F}\) of subgroups \(H\) in \(G\). The kernel of the restriction functor on morphisms is a stable homological ideal in \(KK^G\), and its powers form a filtration \(({\mathcal I}^{\mathcal F}_G)^n\) on \(KK^G\). The results in the paper are formulated in terms of such filtrations. The theory needs injective resolutions, which require infinite products. Therefore, the article works in the larger category of \(\sigma\)-\(C^*\)-algebras throughout. It is shown in an appendix that several results known for \(KK^G\) extend to this larger category. A sample result says that a \(\sigma\)-\(G\)-\(C^*\)-algebra is \(G\)-equivariantly \(KK\)-equivalent to~\(0\) once it is \(H\)-equivariantly \(KK\)-equivalent to~\(0\) for all cyclic subgroups \(H\) in \(G\). This is the generalisation of McClure's restriction map theorem, and it is a consequence of a theorem saying that \(({\mathcal I}^{\mathcal Z}_G)^n=0\) for some \(n>0\), where \({\mathcal Z}\) is the family of cyclic subgroups. As a consequence, in the study of the permanence properties of the Baum--Connes conjecture and related questions for group extensions, one may replace compact subgroups in the quotient by compact cyclic subgroups. Another important theorem compares the completion of \(KK^G(A,B)\) at another decreasing chain of ideals \((I^{\mathcal F}_G)^n\) with the group of maps \(A\to B\) in the localisation of \(KK^G\) at the class of objects that are \(H\)-equivariantly \(KK\)-equivalent to~\(0\) for all \(H\in{\mathcal F}\). This group is further identified with the Kasparov group \(RKK^G(E_{\mathcal F} G,A,B)\), where \(E_{\mathcal F} G\) is the universal \(G\)-CW-complex with isotropy groups in~\({\mathcal F}\).