In this paper, we develop a quantitative -theory for filtered -algebras. Particularly interesting examples of filtered -algebras include group -algebras, crossed product -algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative -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 -théorie quantitative pour les -algèbres filtrées. Parmi les exemples les plus intéressants de telles -algèbres figurent les algèbres de Roe, les -algèbres de groupes et les -algèbres de produits croisés. Nous établissons une version quantitative de la suite exacte à six termes en -théorie ainsi que de la périodicité de Bott. Nous formulons en utilisant la -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.
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
@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},
pages = {605--674},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {65},
number = {2},
year = {2015},
doi = {10.5802/aif.2940},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2940/}
}
TY - JOUR AU - Oyono-Oyono, Hervé AU - Yu, Guoliang TI - On quantitative operator $K$-theory JO - Annales de l'Institut Fourier PY - 2015 SP - 605 EP - 674 VL - 65 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2940/ DO - 10.5802/aif.2940 LA - en ID - AIF_2015__65_2_605_0 ER -
%0 Journal Article %A Oyono-Oyono, Hervé %A Yu, Guoliang %T On quantitative operator $K$-theory %J Annales de l'Institut Fourier %D 2015 %P 605-674 %V 65 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2940/ %R 10.5802/aif.2940 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2940/
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