[-théorie quantitative pour les algèbres d’opérateurs]
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.
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.
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
@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, Tome 65 (2015) no. 2, pp. 605-674. doi : 10.5802/aif.2940. https://aif.centre-mersenne.org/articles/10.5802/aif.2940/
[1] Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Soc. Math. France, Paris, 1976, p. 43-72. Astérisque, No. 32-33 | MR
[2] Classifying space for proper actions and -theory of group -algebras, -algebras: 1943–1993 (San Antonio, TX, 1993) (Contemp. Math.), Volume 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291 | DOI | MR | Zbl
[3] A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980) (Proc. Sympos. Pure Math.), Volume 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521-628 | Zbl
[4] The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 6, pp. 1139-1183 | DOI | MR | Zbl
[5] Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, Volume 29 (1990) no. 3, pp. 345-388 | DOI | MR | Zbl
[6] -theoretic amenability for discrete groups, J. Reine Angew. Math., Volume 344 (1983), pp. 180-195 | DOI | MR | Zbl
[7] Geometrization of the strong Novikov conjecture for residually finite groups, J. Reine Angew. Math., Volume 621 (2008), pp. 159-189 | DOI | MR | Zbl
[8] -theory and -theory for groups which act properly and isometrically on Hilbert space, Invent. Math., Volume 144 (2001) no. 1, pp. 23-74 | DOI | MR | Zbl
[9] A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc., Volume 114 (1993) no. 1, pp. 85-97 | DOI | MR | Zbl
[10] -theoretic amenability for , and the action on the associated tree, J. Funct. Anal., Volume 58 (1984) no. 2, pp. 194-215 | DOI | MR | Zbl
[11] Equivariant -theory and the Novikov conjecture, Invent. Math., Volume 91 (1988) no. 1, pp. 147-201 | DOI | MR | Zbl
[12] -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math., Volume 149 (2002) no. 1, pp. 1-95 | DOI | MR | Zbl
[13] Homotopy invariance of higher signatures and 3-manifold groups, Bull. Soc. Math. France, Volume 136 (2008) no. 1, pp. 1-25 | Numdam | MR | Zbl
[14] -theory for the maximal Roe algebra of certain expanders, J. Funct. Anal., Volume 257 (2009) no. 10, pp. 3239-3292 | DOI | MR | Zbl
[15] Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc., Volume 104 (1993) no. 497, pp. x+90 | DOI | MR | Zbl
[16] The coarse Baum-Connes conjecture and groupoids, Topology, Volume 41 (2002) no. 4, pp. 807-834 | DOI | MR | Zbl
[17] La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory, Volume 17 (1999) no. 3, pp. 215-264 | DOI | MR | Zbl
[18] -theory and -algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993, pp. xii+370 (A friendly approach) | MR | Zbl
[19] The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2), Volume 147 (1998) no. 2, pp. 325-355 | DOI | MR | Zbl
[20] The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., Volume 139 (2000) no. 1, pp. 201-240 | DOI | MR | Zbl
Cité par Sources :
