Bivariant \(K\)-theory and the Novikov conjecture
It is well known that the injectivity of the Baum-Connes assembly map implies the Novikov conjecture. In the paper under review the author used some results of Yu and of some of his previous results to show that if there is a topological amenable action of a countable discrete group \(G\) on some compact Hausdorff space, then for every separable \(G\)-C*-algebra \(A\), the Baum-Connes assembly map \[ \mu: KK^G(\mathcal {EG}, A) \to KK(\mathbb C, C^*_r(G,A)) \] is split injective (Theorem 1.1). The author also proved (Theorem 1.2) a version of the Carlson-Pedersen theorem adapted to the Baum-Connes assembly map: For a countable discrete group \(G\) if the classifying space \(\mathcal {EG}\) is \(G\)-compact and admits a metrizable compactification \(\overline{\mathcal {EG}}\) which is a \(G\)-space and \(H\)-equivariantly contractible for every finite subgroup \(H\) of \(G\) and if \(\lim_{g \to \infty} \text{diam}_{\overline{\mathcal{EG}}} (gK) = 0\) for every compact subset \(K\) of \(\mathcal {EG}\) then the Baum-Connes assembly map is injective.
scientific article; zbMATH DE number 1525495