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

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