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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.

46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
Zbl 0830.46061
Full Text: DOI
[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.
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