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The coarse Baum-Connes conjecture and groupoids. (English) Zbl 1033.19003

Let \(X\) be a discrete metric space with bounded geometry, i.e., such that for every \(R>0\) every ball of radius \(R\) has at most \(N(R)\) elements for some real number \(N(R)>0\). In this article, to every such space \(X\), a locally compact groupoid \(G(X)\) is associated. So, the coarse assembly map for a metric space \(X\) (in the sense of H. Higson and J. Roe [Lond. Math. Soc. Lect. Note Ser. 227, 227–254 (1995; Zbl 0957.58019)]; J. Roe, “Index theory, coarse geometry, and topology of manifolds”, Regional Conf. Ser. Math. 90 (1996; Zbl 0853.58003); G.Yu \(K\)-Theory 9, 199–221 (1995; Zbl 0829.19004)]) is identified with the Baum-Connes assembly map for the groupoid \(G(X)\) with coefficients in the \(C^*\)-algebra \(l^\infty (X,{\mathcal K})\) \[ K^{\text{top}} \bigl(G(X); l^\infty(X: {\mathcal K})\bigr)\to C^ *\bigl( l^\infty (X;{\mathcal K})\ltimes_r G(X)\bigr), \] In the second half of this paper, these results are applied to spaces which admit a uniform embedding into Hilbert space and a new proof of Yu’s result [G. Yu, Invent. Math. 139, 201–240 (2000; Zbl 0956.19004)] that if \(X\) admits a uniform embedding into Hilbert space, the coarse assembly map is a isomorphism is obtained by observing that \(X\) admits a uniform embedding into Hilbert space iff \(G(X)\) admits a proper action on a continuous field of affine Hilbert spaces and using the fact from J.-L. Tu [\(K\)-Theory 17, 303–318 (1999; Zbl 0939.19002)] that for such a groupoid, the Baum-Connes map with coefficients is an isomorphism.
If furthermore \(X\) is a discrete group \(\Gamma\) with a translation-invariant metric, the authors show, using Higson’s descent technique, that \(\Gamma\) also satisfies the Novikov conjecture.

MSC:

19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J22 Exotic index theories on manifolds
22A22 Topological groupoids (including differentiable and Lie groupoids)
46L85 Noncommutative topology
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