On quantitative operator K-theory
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, p. 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 : https://doi.org/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
@article{AIF_2015__65_2_605_0,
     author = {Oyono-Oyono, Herv\'e and Yu, Guoliang},
     title = {On quantitative operator $K$-theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     pages = {605-674},
     doi = {10.5802/aif.2940},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_2_605_0}
}
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/item/AIF_2015__65_2_605_0/

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