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.
Accepted:
Published online:
Classification: 19K35, 46L80, 58J22
Keywords: Baum-Connes Conjecture, Coarse Geometry, Group and Crossed product -algebras, Novikov Conjecture, Operator Algebra -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}, 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 TI - On quantitative operator $K$-theory JO - Annales de l'Institut Fourier PY - 2015 DA - 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/ UR - https://doi.org/10.5802/aif.2940 DO - 10.5802/aif.2940 LA - en ID - AIF_2015__65_2_605_0 ER -
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/
[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: 420729
[2] Classifying space for proper actions and -theory of group -algebras, -algebras: 1943–1993 (San Antonio, TX, 1993) (Contemp. Math.) Tome 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291 | Article | MR: 1292018 | Zbl: 0830.46061
[3] A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980) (Proc. Sympos. Pure Math.) Tome 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 521-628 | Zbl: 0531.57023
[4] The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci., Tome 20 (1984) no. 6, pp. 1139-1183 | Article | MR: 775126 | Zbl: 0575.58030
[5] Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, Tome 29 (1990) no. 3, pp. 345-388 | Article | MR: 1066176 | Zbl: 0759.58047
[6] -theoretic amenability for discrete groups, J. Reine Angew. Math., Tome 344 (1983), pp. 180-195 | Article | MR: 716254 | Zbl: 0511.46066
[7] Geometrization of the strong Novikov conjecture for residually finite groups, J. Reine Angew. Math., Tome 621 (2008), pp. 159-189 | Article | MR: 2431253 | Zbl: 1154.46042
[8] -theory and -theory for groups which act properly and isometrically on Hilbert space, Invent. Math., Tome 144 (2001) no. 1, pp. 23-74 | Article | MR: 1821144 | Zbl: 0988.19003
[9] A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc., Tome 114 (1993) no. 1, pp. 85-97 | Article | MR: 1219916 | Zbl: 0792.55001
[10] -theoretic amenability for , and the action on the associated tree, J. Funct. Anal., Tome 58 (1984) no. 2, pp. 194-215 | Article | MR: 757995 | Zbl: 0559.46030
[11] Equivariant -theory and the Novikov conjecture, Invent. Math., Tome 91 (1988) no. 1, pp. 147-201 | Article | MR: 918241 | Zbl: 0647.46053
[12] -théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math., Tome 149 (2002) no. 1, pp. 1-95 | Article | MR: 1914617 | Zbl: 1084.19003
[13] Homotopy invariance of higher signatures and 3-manifold groups, Bull. Soc. Math. France, Tome 136 (2008) no. 1, pp. 1-25 | Numdam | MR: 2415334 | Zbl: 1179.19004
[14] -theory for the maximal Roe algebra of certain expanders, J. Funct. Anal., Tome 257 (2009) no. 10, pp. 3239-3292 | Article | MR: 2568691 | Zbl: 1185.46047
[15] Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc., Tome 104 (1993) no. 497, x+90 pages | Article | MR: 1147350 | Zbl: 0780.58043
[16] The coarse Baum-Connes conjecture and groupoids, Topology, Tome 41 (2002) no. 4, pp. 807-834 | Article | MR: 1905840 | Zbl: 1033.19003
[17] La conjecture de Baum-Connes pour les feuilletages moyennables, -Theory, Tome 17 (1999) no. 3, pp. 215-264 | Article | MR: 1703305 | Zbl: 0939.19001
[18] -theory and -algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993, xii+370 pages (A friendly approach) | MR: 1222415 | Zbl: 0780.46038
[19] The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2), Tome 147 (1998) no. 2, pp. 325-355 | Article | MR: 1626745 | Zbl: 0911.19001
[20] The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., Tome 139 (2000) no. 1, pp. 201-240 | Article | MR: 1728880 | Zbl: 0956.19004
Cited by Sources: