no. 2
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.

Received:
Accepted:
Published online:
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 
@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  - 
%0 Journal Article
%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://doi.org/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/

[1] Atiyah, M. F. 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] Baum, Paul; Connes, Alain; Higson, Nigel Classifying space for proper actions and K-theory of group C * -algebras, C * -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] Connes, A. 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] Connes, A.; Skandalis, G. 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] Connes, Alain; Moscovici, Henri Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, Tome 29 (1990) no. 3, pp. 345-388 | Article | MR: 1066176 | Zbl: 0759.58047

[6] Cuntz, Joachim K-theoretic amenability for discrete groups, J. Reine Angew. Math., Tome 344 (1983), pp. 180-195 | Article | MR: 716254 | Zbl: 0511.46066

[7] Gong, Guihua; Wang, Qin; Yu, Guoliang 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] Higson, Nigel; Kasparov, Gennadi E-theory and KK-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] Higson, Nigel; Roe, John; Yu, Guoliang 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] Julg, Pierre; Valette, Alain K-theoretic amenability for SL 2 (Q p ), 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] Kasparov, G. G. Equivariant KK-theory and the Novikov conjecture, Invent. Math., Tome 91 (1988) no. 1, pp. 147-201 | Article | MR: 918241 | Zbl: 0647.46053

[12] Lafforgue, Vincent K-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] Matthey, Michel; Oyono-Oyono, Hervé; Pitsch, Wolfgang 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] Oyono-Oyono, Hervé; Yu, Guoliang K-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] Roe, John 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] Skandalis, G.; Tu, J. L.; Yu, G. The coarse Baum-Connes conjecture and groupoids, Topology, Tome 41 (2002) no. 4, pp. 807-834 | Article | MR: 1905840 | Zbl: 1033.19003

[17] Tu, Jean-Louis La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory, Tome 17 (1999) no. 3, pp. 215-264 | Article | MR: 1703305 | Zbl: 0939.19001

[18] Wegge-Olsen, N. E. K-theory and C * -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] Yu, Guoliang 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] Yu, Guoliang 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: