Direct limits in an equivariant K theory defined by proper cocycles (Q1825416)

Related to the Baum-Connes K-theory of the reduced crossed product \(C^*\)-algebras arising from smooth and proper action of a Lie group G on a smooth manifold X, the author gives here, after a full discussion of the definition of \(K^*(G,X)\) by using vector bundles with finite dimensional fibers, a result concerning the behavior of \(K^*(G,X)\) under certain direct limits, of the type \(G=_{i}G_ i\) where \(G_ i\) are the Lie subgroups and the indices i are elements of the positive integers with their usual ordering. Namely, if each \(G_ i\) is an open and closed subgroup of G and \(i_ 1<i_ 2\) implies \(G_{i_ 1}\subset G_{i_ 2}\), the author proves that \(_{i}K^*(G_ i,X)\) makes sense and is equal to \(K^*(G,X)\). So, an analogous result there is in the K- theory of \(C^*\)-algebras: \(K_*(C^*(G,C_ 0(X)))=_{i}K_*(C^*(G_ i,C_ 0(X)))\) such that the isomorphism of \(K^*(G,K)\) and \(K_*(C^*(G,C_ 0(X)))\) the existence of which is conjectured by Baum and Connes reduces to analogous isomorphisms involving the \(G_ i.\) The existence of this isomorphism is proved if each \(G_ i\) is compact and for any abelian Lie group.