×

zbMATH — the first resource for mathematics

Baum-Connes conjecture for some semi-direct products. (English) Zbl 0959.46049
Following a general construction suggested by P. Baum, A. Connes and N. Higson [Contemp. Math. 167, 241-291 (1994; Zbl 0830.46061)] and using the Dirac/dual Dirac method that G. Kasparov designed to prove special cases of the Novikov conjecture the author proves the Baum-Connes conjecture for the reduced \(C^\ast\)-algebra of a second countable locally compact group \(G\) in the following case. The group \(G\) is assumed to have a \(\gamma\)-element to make the Kasparov method work. Moreover, \(G\) is a semi-direct product of a group \(H\), for which the Baum-Connes conjecture holds, and an amenable group N. Now the conjecture for G with coefficients in a \(G\)-algebra \(B\) can be reduced to the conjecture for \(H\) with coefficients in the crossed product of \(B\) by \(N\) provided that \(H\) is almost connected or has a compact-open subgroup.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
Citations:
Zbl 0830.46061
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Baaj and G. Skandalis, C-algeAbres deHopf et theAorie de Kasparov eAquivariante, K-theory 2 (1989), 683-721.
[2] Baum P., Contemp. Math. 167 pp 241– (1994)
[3] N. Bourbaki, InteAgration, Chapitres VII-VIII, Herman, Paris1963.
[4] J. Dixmier, Les C-algeAbreset leurs repreAsentations, Gauthier-Villars, Paris 1964.
[5] F. P. Greenleaf, Invariant means on topological groups, Van Nostrand, New York 1969. · Zbl 0174.19001
[6] E. Guenter, N. Higson, and J. Trout, Equivariant E-theory for C-algebras, preprint 1997.
[7] N. Higson and G. Kasparov, OperatorK-theory for groups which act properly and isometrically on Euclidean space, preprint 1997. · Zbl 0888.46046
[8] Julg P., Austral. Nat. Univ. 16 pp 143– (1988)
[9] G. Kasparov, EquivariantKK-theory and the Novikov conjecture, Invent. math. 91 (1988), 147-201. · Zbl 0647.46053
[10] G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, preprint 1998. · Zbl 1029.19003
[11] G. Kasparov and G. Skandalis, Groups acting on buildings, K-theory 4 (1991), 303-337. · Zbl 0738.46035
[12] H. Oyono, La conjecture de Baum-Connes pour les groupes agissant sur des arbres, PhD thesis, UniversiteA Claude Bernard Lyon I, 1997. · Zbl 0918.46062
[13] Ann. Math. 73 (2) pp 295– (1961)
[14] G. K. Pedersen, C-algebrasand their automorphism groups, The London Mathematical Society, Academic Press, London 1979.
[15] J. L. Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, preAprint 1998. · Zbl 0939.19001
[16] J. L. Tu, The Baum-Connes conjecture and discrete group actions on trees, preAprint aA l’UniversiteA de Paris VII, 1997.
[17] J. L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, PhD thesis, UniversiteA Paris VII, 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.