Abstract
The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group C*-algebras.
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References
P. Baum and A. ConnesK-theory for Lie groups and foliations, Preprint, (1982).
P. Baum, A. Connes and N. Higson, 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. (1994), 240–291.
P. Baum, N. Higson and R. Plymen, A proof of the Baum-Connes conjecture for p-adic GL(n),C. R. Acad. Sci. Paris Sér. I, 325, (1997), 171–176.
J.-B. Bost, Principe d’Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math., 101, (1990), 261–333.
J. ChabertBaum-Connes conjecture for some semi-direct products, J. Reine Angew. Math., 521, (2000), 161–184.
J. Chabert and S. Echterhoff, Permanence properties of the Baume-Connes conjecture,Documenta Math. 6, (2001), 127–183.
A. Connes, Non-commutative geometry, Academic Press, (1994).
J. Cuntz, K-theoretic amenability for discrete groups,J. Reine Angew. Math., 344, (1983), 180–195.
J. Cuntz, Bivariante K-theorie für lokalconvexe Algebren und der Chern-Connes Character, Doc. Math., 2, (1997), 139–182.
Novikov conjectures, Index theorems and Rigidity, Edited by S. Ferry, A. Ranicki and J. Rosenberg, Volume 1, London Mathematical Society, LNS 226, (1993).
M. Gromov, Spaces and questions, Preprint (1999).
U. Haagerup, An example of a nonnuclear C* -algebra which has the metric approximation property, Inv. Math., 50, (1979), 279–293.
P. de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolis-saint, C. R. Acad. Sci. Paris Sér. I, 307, (1988), 771–774.
P. de la Harpe and A. ValetteLa propriété (T) de Kazdhan pour les groupes localement compacts, Astérisque, (1989).
N. Higson and G. KasparovOperator K-theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc., 3, (1997), 131–142.
N. Higson, The Baum-Connes conjecture,Proc. of the Int. Cong. of Math., Vol. II (Berlin, 1998), Doc. Math., (1998), 637–646.
N. Higson, Bivariant K-theory and the Novikov conjecture, Preprint, (1999).
P. Jolissaint, Rapidly decreasing functions in reduced C* -algebra of groups, Trans. Amer. Math. Soc., 317, (1990), 167–196.
P. Julg and A. Valette, K-theoretic amenability for SL 2 (Q z ,),and the action on the associated tree, J. Funct. Anal, 58, (1984), 194–215.
P. Julg and G. KasparovOperator K-theory for the group SU(n, 1), J. Reine Angew. Math., 463 (1995), 99–152.
P. Julg Remarks on the Baum-Connes conjecture and Kazhdan’s property T, Operator algebras and their applications, Waterloo (1994/1995), Fields Inst. Commun., Amer. Math. Soc., 13 (1997), 145–153.
P. Julg, Travaux de N. Higson et G. Kasparov sur la conjecture de Baum-Connes,Séminaire Bourbaki. Vol. 1997/98, Astérisque, 252 (1998), No. 841, 4, 151–183.
G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology, Izv. Akad. Nauk SSSR Ser. Mat., 39 no. 4, (1975), 796–838.
G. G. Kasparov, The operator K-functor and extensions of C* -algebras, Math. USSR Izv., 16 (1980), 513–572.
G. G. Kasparov, Operator K-theory and its applications: elliptic operators, group representations, higher signatures,Cam-extensions, Proc. of the Int. Cong. of Math., (Warsaw, 1983), 987–1000.
G. G. Kasparov, K-theory, group C* -algebras, and higher signatures (conspectus), Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), 101–146, London Math. Soc. LNS 226, (1981).
G. G. Kasparov, Lorentz groups: K-theory of unitary representations and crossed products, Dokl. Akad. Nauk SSSR, 275 (1984), 541–545.
G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91 (1988), 147–201.
G. G. Kasparov and G. Skandalis, Groups acting on buildings,operator K-theory, and Novikov’s conjecture, K-Theory, 4 (1991), 303–337.
G. Kasparov and G. Skandalis, Groupes boliques et conjecture de Novikov Comptes Rendus Acad. Sc., 319 (1994), 815–820.
V. Lafforgue, Une démonstration de la conjecture de Baum-Connes pour les groupes réductifs sur un corps p-adique et pour certains groupes discrets possédant la propriété (T), C. R. Acad. Sci. Paris Sér. I, 327 (1998), 439–444.
V. Lafforgue Compléments à la démonstration de la conjecture de Baum-Connes pour certains groupes possédant la propriété (T), C. R. Acad. Sci. Paris Sér. I, 328 (1999), 203–208.
V. Lafforgue KK-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Thèse de Doctorat de l’université Paris 11 (mars 1999).
V. Lafforgue, A proof of property (RD) for cocompact lattices in SL 3 (I“.) and SL 3 (C), Journal of Lie Theory 10, (2000), 255–267.
P.-Y. Le Gall, Théorie de Kasparov équivariante et groupoïdes, C. R. Acad. Sci. Paris Sér. I, 324 (1997), 695–698.
P.-Y. Le Gall, Théorie de Kasparov équivariante et groupoïdes. I, K-Theory, 16 (1999), 361–390.
A. S. Mishchenko, Homotopy invariants of multiply connected manifolds. I. Rational invariants, Math. USSR Izv., 4 (1970), 509–519, translated from Izv. Akad. Nauk SSSR Ser. Mat., 34, (1970), 501–514.
A. S. Mishchenko, Infinite-dimensional representations of discrete groups,and higher signatures, Math. USSR Izv., 8 no. 1, (1974), 85–111.
H. Oyono-Oyono, La conjecture de Baum—Connes pour les groupes agissant sur les arbres, C. R. Acad. Sci. Paris Sér. I Math., 326, (1998), 799–804.
H. Oyono-Oyono, Baum—Connes conjecture and group actions on trees, Preprint.
M. V. Pimsner, KK-groups of crossed products by groups acting on trees, Invent. Math., 86, (1986), 603–634.
J. Ramagge, G. Robertson and T. Steger, A Haagerup inequality for A l x A l and A2 buildings, Geom. Funct. Anal., 8, (1998), 702–731.
G. Skandalis, Kasporov’s bivariant K-theory and applications, Expositiones Math., 9 (1991), 193–250.
G. Skandalis, Progrès récents sur la conjecture de Baum—Connes, contribution de Vincent Lafforgue, Séminaire Bourbaki, No. 869 (novembre 1999).
J.-L. Tu, The Baum—Connes conjecture and discrete group actions on trees, K-Theory, 17 (1999), 303–318.
J.-L. Tu, La conjecture de Baum—Connes pour les feuilletages moyennables, K-Theory, 17 (1999), 215–264.
J.-L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory, 16 (1999), 129–184.
A. Valette, Introduction to the Baum—Connes conjecture, to appear in the Series “Lectures in Mathematics — ETH Zürich”, Birkhäuser.
A. Wassermann, Une démonstration de la conjecture de Connes—Kasparov pour les groupes de Lie linéaires connexes réductifs, C. R. Acad. Sci. Paris Sér. I, 304, (1987), 559–562.
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Lafforgue, V. (2001). Banach KK-Theory and the Baum-Connes Conjecture. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_4
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