Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T06:13:20.645Z Has data issue: false hasContentIssue false
  • English
  • Français

The Baum-Connes conjecture for KK-theory

Published online by Cambridge University Press:  07 April 2010

Otgonbayar Uuye
Otgonbayar Uuye
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Denmark, otogo@math.ku.dk
Get access

Abstract

We define and compare two bivariant generalizations of the topological K - group Ktop(G). We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.

Keywords

Kasparov KK-theoryBaum-Connes ConjectureUCT
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 1 , August 2011 , pp. 3 - 29
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abe75.Abels, Herbert, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann. 212 (1974/75), 1–19. MR MR0375264 (51 #11460)CrossRefGoogle Scholar
BCH94.Baum, Paul, Connes, Alain and Higson, Nigel, Classifying space for proper actions and K-theory of group C*-algebras, C*-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR MR1292018 (96c:46070)Google Scholar
Bil04.Biller, Harald, Characterizations of proper actions, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2,429–439. MR MR2040583 (2004k:57043)CrossRefGoogle Scholar
Bla98.Blackadar, Bruce, K-theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications 5 , Cambridge University Press, Cambridge, 1998. MR MR1656031 (99g:46104)Google Scholar
BMP03.Baum, Paul, Millington, Stephen and Plymen, Roger, Local-global principle for the Baum-Connes conjecture with coefficients, K-Theory 28 (2003), no. 1, 1–18. MR MR1988816 (2004d:46085)CrossRefGoogle Scholar
CE01a.Chabert, Jérôme and Echterhoff, Siegfried, Permanence properties of the Baum-Connes conjecture, Doc. Math. 6 (2001), 127–183 (electronic). MR MR1836047 (2002h:46117)CrossRefGoogle Scholar
CE01b.Chabert, Jérôme and Echterhoff, Siegfried, Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions, K-Theory 23 (2001), no. 2, 157–200. MR MR1857079 (2002m:19003)CrossRefGoogle Scholar
CEN03.Chabert, Jérôme, Echterhoff, Siegfried and Nest, Ryszard, The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups, Publ. Math. Inst. Hautes Études Sci. (2003), no. 97, 239–278. MR MR2010742 (2004j:19004)CrossRefGoogle Scholar
CEOO03.Chabert, Jérôme, Echterhoff, Siegfried and Oyono-Oyono, Hervé, Shapiro's lemma for topological K-theory of groups, Comment. Math. Helv. 78 (2003), no. 1, 203–225. MR MR1966758 (2004c: 19005)CrossRefGoogle Scholar
CEOO04.Chabert, J., Echterhoff, S. and Oyono-Oyono, H., Going-down functors, the Künneth formula and the Baum-Connes conjecture, Geom. Funct. Anal. 14 (2004), no. 3, 491–528. MR MR2100669 (2005h: 19005)CrossRefGoogle Scholar
Con94.Connes, Alain, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994. MR MR1303779 (95j:46063)Google Scholar
DL96.Dadarlat, Marius and Loring, Terry A., A universal multicoefficient theorem for the Kasparov groups, Duke Math. J. 84 (1996), no. 2, 355–377. MR MR1404333 (97f:46109)CrossRefGoogle Scholar
HK01.Higson, Nigel and Kasparov, Gennadi, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74. MR MR1821144 (2002k:19005)CrossRefGoogle Scholar
HLS02.Higson, N., Lafforgue, V. and Skandalis, G., Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354. MR MR1911663(2003g:19007)CrossRefGoogle Scholar
Jul02.Julg, Pierre, La conjecture de Baum-Connes à coefficients pour le groupe Sp(n, 1), C. R. Math. Acad. Sci. Paris 334 (2002), no. 7, 533–538. MR MR1903759 (2003d:19007)CrossRefGoogle Scholar
Kas88.Kasparov, G. G., Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR MR918241 (88j:58123)CrossRefGoogle Scholar
KS91.Kasparov, G. G. and Skandalis, G., Groups acting on buildings, operator K-theory and Novikov's conjecture, K-Theory 4 (1991), no. 4, 303–337. MR MR1115824 (92h:19009)CrossRefGoogle Scholar
KS03.Kasparov, Gennadi and Skandalis, Georges, Groups acting properly on “bolic” spaces and the Novikov conjecture, Ann. of Math. (2) 158 (2003), no. 1, 165–206. MR MR1998480 (2004j:58023)CrossRefGoogle Scholar
MN06.Meyer, Ralf and Nest, Ryszard, The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259. MR MR2193334 (2006k: 19013)CrossRefGoogle Scholar
RS87.Rosenberg, Jonathan and Schochet, Claude, The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor, Duke Math. J. 55 (1987), no. 2, 431–474. MR MR894590 (88i:46091)CrossRefGoogle Scholar
Ska88.Skandalis, Georges, Une notion de nucléarité en K-théorie (d'après J. Cuntz), K-Theory 1 (1988), no. 6, 549–573. MR MR953916 (90b:46131)CrossRefGoogle Scholar
Tak67.Takesaki, Masamichi, Covariant representations of C*-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303. MR MR0225179 (37 #774)CrossRefGoogle Scholar
Tu99.Tu, Jean Louis, La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory 16 (1999), no. 2, 129–184. MR MR1671260 (99m:46163)CrossRefGoogle Scholar