Counterexamples to the Baum-Connes conjecture. (English) Zbl 1014.46043

Higson, N.; Lafforgue, V.; Skandalis, G.

Geom. Funct. Anal. 12, No. 2, 330-354 (2002).

To an action of a group \(G\) on a \(C^*\)-algebra \(A\) one can associate topological \(K\)-theory groups \(K^{\text{top}}_*(G;A)\) and operator \(K\)-theory groups \(K_*(A\rtimes_r G)\), which are related by the assembly map \(\mu_A:K^{\text{top}}_*(G;A)\to K_*(A\rtimes_r G)\). The Baum-Connes conjecture with coefficients asserts that \(\mu_A\) is an isomorphism. It is a generalization of the classical Baum-Connes conjecture, which asserts the same for \(A={\mathbb C}\). Another generalization of the conjecture uses groupoids instead of groups.
The authors provide counterexamples to: injectivity and surjectivity of the assembly map for Hausdorff groupoids; injectivity and surjectivity of the assembly map for holonomy groupoids of foliations; surjectivity of the assembly map for coarse geometric spaces; surjectivity of the assembly map for discrete group actions on commutative \(C^*\)-algebras.
The main idea is that the failure of exactness, when short exact sequences of algebras are completed to reduced \(C^*\)-algebras, can be detected at the level of \(K\)-theory, where the Baum-Connes conjecture predicts exactness.

Reviewer: V. M. Manuilov (Moskva)

MSC:

46L80

\(K\)-theory and operator algebras (including cyclic theory)

Keywords:

group \(C^*\)-algebra; Baum-Connes conjecture; \(K\)-theory; assembly map