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Morphisme de Baum-Connes tordu par une représentation non unitaire

Published online by Cambridge University Press:  08 December 2009

Maria Paula Gomez-Aparicio
Maria Paula Gomez-Aparicio
Affiliation:
Institut de Mathématiques de Jussieu, Projet d'algèbres d'Opérateurs et représentations, 175 rue du chevaleret, 75013 Paris, France. [email protected]
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Abstract

Let G be a locally compact group and ρ a non-unitary finite dimensional representation of G. We consider tensor products of ρ by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.

Soit G un groupe localement compact et ρ une représentation de dimension finie de G non unitaire. On définit des algèbres de Banach analogues aux C*-algèbres de groupe, C*(G) et C*r(G), en considérant l'ensemble des représentations de la forme ρ ⊗ π, où π parcourt un ensemble de représentations unitaires de G. On calcule la K-théorie de ces algèbres pour une large classe de groupes vérifiant la conjecture de Baum-Connes.

Keywords

Non-unitary representationsBanach algebrasKK-theoryBaum-Connes conjecture
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 1 , August 2010 , pp. 23 - 68
Copyright
Copyright © ISOPP 2009

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References

Références

BCH94.Baum, P., Connes, A., and Higson, N., Classifying space for proper actions and K-theory of group C*-algebras, C*-algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291Google Scholar
Bos90.Bost, J. B., Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math. 101 (1990), 261–333CrossRefGoogle Scholar
CEM01.Chabert, J., Echterhoff, S., and Meyer, R., Deux remarques sur l'application de Baum-Connes, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 7, 607–610Google Scholar
Cha03.Chatterji, I., Property (RD) for cocompact lattices in a finite product of rank one Lie groups with some rank two Lie groups, Geom. Dedicata 96 (2003), 161–177CrossRefGoogle Scholar
GA07a.Gomez-Aparicio, M. P., Propriété (T) et morphisme de Baum-Connes tordus par une représentation non-unitaire, Ph.D. thesis, Université de Paris VII, décembre 2007Google Scholar
GA07b.Gomez-Aparicio, M. P., Sur la propriété (T) tordue par un produit tensoriel, J. Lie Theory 17 (2007), 505–524Google Scholar
GA08.Gomez-Aparicio, M. P., Représentations non-unitaires, morphisme de Baum-Connes et complétions inconditionnelles, to appear in Journal of Noncommutative Geometry, 2008Google Scholar
HG04.Higson, N. and Guentner, E., Group C*-algebras and K-theory, Noncommutative geometry, Lecture Notes in Math., vol. 1831, Springer, Berlin, 2004, pp. 137–251Google Scholar
HK01.Higson, N. and Kasparov, G. G., E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23–74CrossRefGoogle Scholar
HLS02.Higson, N., Lafforgue, V., and Skandalis, G., Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354CrossRefGoogle Scholar
JK95.Julg, P. and Kasparov, G., Operator K-theory for the group SU(n,1), J. Reine Angew. Math. 463 (1995), 99–152 (English)Google Scholar
Jul97.Julg, P., Remarks on the Baum-Connes conjecture and Kazhdan's property T, Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 145–153Google Scholar
Kas84.Kasparov, G. G., Lorentz groups: K-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR 275 (1984), no. 3, 541–545Google Scholar
Kas88.Kasparov, G. G., Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201CrossRefGoogle Scholar
Kaz67.Kazhdan, D. A., On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74Google 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, 303–337CrossRefGoogle Scholar
KS03.Kasparov, G. G. and Skandalis, G., Groups acting properly on “bolic” spaces and the Novikov conjecture, Ann. of Math. (2) 158 (2003), no. 1, 165–206Google Scholar
Laf00.Lafforgue, V., A proof of property (RD) for cocompact lattices of SL(3,R) and SL(3,C), J. Lie Theory 10 (2000), no. 2, 255–267Google Scholar
Laf02a.Lafforgue, V., Banach KK-theory and the Baum-Connes conjecture, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 795–812Google Scholar
Laf02b.Lafforgue, V., K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002), no. 1, 1–95Google Scholar
LG97.Le Gall, P. Y., Théorie de Kasparov équivariante et groupoïdes, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 6, 695–698Google Scholar
Par06.Paravicini, W., KK-theory for Banach algebras and proper groupoids, Ph.D. thesis, November 2006Google Scholar
Ska91.Skandalis, G., Kasparov's bivariant K-theory and applications, Expo. Math. 9 (1991), no. 3, 193–250Google Scholar
Tu99.Tu, J. L., La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory 16 (1999), no. 2, 129–184CrossRefGoogle Scholar
Tza00.Tzanev, K., C*-algèbres de Hecke et K-théorie, Ph.D. thesis, Université de Paris VII, 2000.Google Scholar