K-theory for the maximal Roe algebra of certain expanders. (English) Zbl 1185.46047
Summary: We study the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 20F65 | Geometric group theory |
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
Keywords:
Baum-Connes conjecture; coarse geometry; expanders; Novikov conjecture; operator algebra K-theory; Roe algebrasReferences:
| [1] | Baum, Paul; Connes, Alain, K-theory for discrete groups, (Operator Algebras and Applications (1989), Cambridge University Press), 1-20 · Zbl 0685.46041 |
| [2] | Chabert, Jérôme; Echterhoff, Siegfried, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6, 127-183 (2001), (electronic) · Zbl 0984.46047 |
| [3] | Chen, Xiaoman; Tessera, Romain; Wang, Xianjin; Yu, Guoliang, Metric sparsification and operator norm localization, Adv. Math., 218, 1496-1511 (2008) · Zbl 1148.46040 |
| [4] | Gong, Guihua; Wang, Qin; Yu, Guoliang, Geometrization of the strong Novikov conjecture for residually finite groups, J. Reine Angew. Math., 621, 159-189 (2008) · Zbl 1154.46042 |
| [5] | Gromov, Misha, Spaces and questions, Geom. Funct. Anal., I, 2, 118-161 (2000) · Zbl 1006.53035 |
| [6] | Erik Guentner, Romain Tessera, Guoliang Yu, Operator norm localization for linear groups and its applications to K-theory, preprint, 2008; Erik Guentner, Romain Tessera, Guoliang Yu, Operator norm localization for linear groups and its applications to K-theory, preprint, 2008 · Zbl 1222.46056 |
| [7] | Higson, Nigel; Lafforgue, Vincent; Skandalis, Georges, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., 12, 2, 330-354 (2002) · Zbl 1014.46043 |
| [8] | Higson, Nigel; Roe, John, Analytic \(K\)-Homology, Oxford Math. Monogr., Oxford Sci. Publ. (2000), Oxford University Press: Oxford University Press Oxford · Zbl 0968.46058 |
| [9] | Higson, Nigel; Roe, John; Yu, Guoliang, A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc., 114, 1, 85-97 (1993) · Zbl 0792.55001 |
| [10] | Lubotzky, Alexander, Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math., vol. 125 (1994), Birkhäuser Verlag: Birkhäuser Verlag Basel, With an appendix by Jonathan D. Rogawski · Zbl 0826.22012 |
| [11] | Mislin, Guido; Valette, Alain, Proper Group Actions and the Baum-Connes Conjecture, Adv. Courses Math. CRM Barcelona (2003), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1028.46001 |
| [12] | Oyono-Oyono, Hervé, Baum-Connes conjecture and group actions on trees, \(K\)-Theory, 24, 2, 115-134 (2001) · Zbl 1008.19001 |
| [13] | Roe, John, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc., 104, 497 (1993) · Zbl 0780.58043 |
| [14] | Skandalis, Georges; Tu, Jean-Louis; Yu, Guoliang, The coarse Baum-Connes conjecture and groupoids, Topology, 41, 4, 807-834 (2002) · Zbl 1033.19003 |
| [15] | Tu, Jean-Louis, La conjecture de Baum-Connes pour les feuilletages moyennables, \(K\)-Theory, 17, 3, 215-264 (1999) · Zbl 0939.19001 |
| [16] | Kroum Tzanev, K-théorie des \(C^{\ast;}\)-algèbres de groupes, Théorème de l’indice, Géométrie co-uniforme, PhD thesis, Université Paris 7-Denis Diderot, 2000; Kroum Tzanev, K-théorie des \(C^{\ast;}\)-algèbres de groupes, Théorème de l’indice, Géométrie co-uniforme, PhD thesis, Université Paris 7-Denis Diderot, 2000 |
| [17] | Wegge-Olsen, N. E., \(K\)-Theory and \(C^\ast \)-Algebras, Oxford Sci. Publ. (1993), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, (a friendly approach) · Zbl 0780.46038 |
| [18] | Yu, Guoliang, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139, 201-240 (2000) · Zbl 0956.19004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
