Classifying space for proper actions and \(K\)-theory of group \(C^*\)- algebras. (English) Zbl 0830.46061

Baum, Paul; Connes, Alain; Higson, Nigel

Doran, Robert S. (ed.), \(C^*\)-Algebras: 1943-1993. A fifty year celebration. AMS special session commemorating the first fifty years of \(C^*\)-algebra theory, January 13-14, 1993, San Antonio, TX, USA. Providence, RI: American Mathematical Society. Contemp. Math. 167, 241-291 (1994).

Let \(G\) be a group which is locally compact, Hausdorff and second countable. Denote by \(C^*_r (G)\) the completion in the operator norm of the convolution algebra \(L_1 (G)\), viewed as an algebra of operators on \(L_2 (G)\). The purpose of this article is to describe in some detail Baum-Connes’ conjecture concerning the \(C^*\)-algebra \(K\)- theory groups \(K_j (C^*_r (G))\). The advantage of the new version is that it is simpler and applies more generally than the earlier statement. Using Kasparov’s \(KK\)-theory the authors form the equivariant \(K\)-homology groups \(K^G_j (\underline {E} G)\). A class in \(K^G_j (\underline {E} G)\) is represented by an abstract \(G\)-equivariant elliptic operator on \(\underline {E} G\). Following Kasparov the authors define a homomorphism of \(K\)-homology groups \(\mu: K^G_j (\underline {E} G)\to K_j (C^*_r (G))\) by assigning to each elliptic operator its index. The discussed conjecture is that \(\mu\) is an isomorphism of abelian groups.
For the entire collection see [Zbl 0798.00020].

Reviewer: V.Deundyak (Rostov-na-Donu)

MSC:

46L80	\(K\)-theory and operator algebras (including cyclic theory)
58J22	Exotic index theories on manifolds

Keywords:

convolution algebra; Baum-Connes’ conjecture; \(C^*\)-algebra \(K\)-theory groups; Kasparov’s \(KK\)-theory; equivariant \(K\)-homology groups; elliptic operator; index