On quantitative operator K-theory
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 605-674.

In this paper, we develop a quantitative K-theory for filtered C * -algebras. Particularly interesting examples of filtered C * -algebras include group C * -algebras, crossed product C * -algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative K-theory to formulate a quantitative version of the Baum-Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.

Dans cet article, nous développons une K-théorie quantitative pour les C * -algèbres filtrées. Parmi les exemples les plus intéressants de telles C * -algèbres figurent les algèbres de Roe, les C * -algèbres de groupes et les C * -algèbres de produits croisés. Nous établissons une version quantitative de la suite exacte à six termes en K-théorie ainsi que de la périodicité de Bott. Nous formulons en utilisant la K-théorie quantitative une version quantitative de la conjecture de Baum-Connes. Nous montrons que cette conjecture de Baum-Connes quantitative est vérifiée pour une large classe de groupes.

DOI: 10.5802/aif.2940
Classification: 19K35, 46L80, 58J22
Keywords: Baum-Connes Conjecture, Coarse Geometry, Group and Crossed product $C^*$-algebras, Novikov Conjecture, Operator Algebra $K$-theory, Roe Algebras
Mot clés : Conjecture de Baum-Connes, Géométrie à Grande Echelle, $C^*$-algèbres de Groupes et de Produits Croisés, Conjecture de Novikov, $K$-théorie pour les Algèbres d’Opérateurs
Oyono-Oyono, Hervé 1; Yu, Guoliang 2

1 Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine & CNRS, site du Saulcy, Bâtiment A, 57045 METZ (France)
2 Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368 (USA)
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Oyono-Oyono, Hervé; Yu, Guoliang. On quantitative operator $K$-theory. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 605-674. doi : 10.5802/aif.2940. http://www.numdam.org/articles/10.5802/aif.2940/

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