The coarse Baum–Connes conjecture and groupoids

To every discrete metric space with bounded geometry X we associate a groupoid G(X) for which the coarse assembly map for X is equivalent to the Baum–Connes assembly map for G(X) with coefficients in the C∗-algebra ℓ∞(X,K). We thus obtain a new proof of the fact that if X admits a uniform embedding into Hilbert space, the coarse assembly map is an isomorphism. If furthermore X is a discrete group Γ with a translation-invariant metric, we show, using Higsonʹs descent technique, that Γ also satisfies the Novikov conjecture. This removes the finiteness condition in (Yu, Invent. Math. 139 (2000) 201–204).