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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
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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

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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