Equivariant higher-index problems for proper actions and nonpositively curved manifolds. (English) Zbl 1446.58004
The paper under review proves the injectivity of an equivariant higher index map which generalizes both the (classical) Baum-Connes map and the “coarse” Baum-Connes map introduced by John Roe. Namely, this map is \[\mu^{\Gamma} : \lim_{d \to \infty} K^{\Gamma}_*(P_d(X)) \to K_*(C^*(X)^{\Gamma})\] where \(\Gamma\) is a countable discrete group which acts proerly and isometrically on the metric space \(X\). The assumptions that lead to this result are that \(X\) is discrete with bounded geometry and admits a coarse embedding to a simply connected, complete Riemannian manifold \(M\) with nonpositive sectional curvature.
Reviewer: Iakovos Androulidakis (Athína)
MSC:
| 58B34 | Noncommutative geometry (à la Connes) |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
Keywords:
equivariant Roe algebras; equivariant higher-index map; proper group action; Baum-Connes mapReferences:
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