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The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

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Abstract

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero.

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References

  1. H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann., 212 (1974), 1–19.

    Google Scholar 

  2. M. Atiyah, R. Bott and A. Shapiro, Clifford Modules, Topology 3, Suppl. 1 (1964), 3–38.

  3. P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-theory of group C*-algebras, Contemp. Math., 167 (1994), 241–291.

    Google Scholar 

  4. 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, Math., 325, no. 2 (1997), 171–176.

  5. B. Blackadar, K-theory for operator algebras, MSRI Pub. 5, Springer 1986.

  6. E. Blanchard, Deformations de C*-algebres de Hopf, Bull. Soc. Math. Fr., 124 (1996), 141–215.

    Google Scholar 

  7. J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer 1998.

  8. A. Borel, Linear Algebraic Groups, Springer, GTM 126 (1991).

  9. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485–535.

  10. A. Borel and J. Tits, Groupes reductifs, Inst. Hautes Études Sci., Publ. Math., 27 (1965), 55–150.

  11. J. Chabert and S. Echterhoff, Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions, K-Theory, 23 (2001), 157–200.

  12. J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6 (2001), 127–183.

    Google Scholar 

  13. J. Chabert, S. Echterhoff and R. Meyer, Deux remarques sur la conjecture de Baum-Connes, C. R. Acad. Sci., Paris, Sér. I 332, no 7 (2001), 607–610.

  14. J. Chabert, S. Echterhoff and H. Oyono-Oyono, Going-Down functors, the Künneth formula, and the Baum-Connes conjecture, Preprintreihe SFB 478, Münster.

  15. P.-A. Cherix, M. Cowling, P. Jolisssaint, P. Julg and A. Valette, Groups with the Haagerup property, Progress in Mathematics 197, Birkhäuser 2000.

  16. C. Chevalley, Theorie des groupes de Lie, Groupes algebriques, Theoremes generaux sur les algebres de Lie, 2ieme ed., Hermann & Cie. IX, Paris, 1968.

  17. A. Connes and H. Moscovici, The L2-index theorem for homogeneous spaces of Lie groups, Ann. Math., 115 (1982), 291–330.

    Google Scholar 

  18. J. Dixmier, C*-algebras (English Edition). North Holland Publishing Company 1977.

  19. M. Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math., 149 (1983), 153–213.

    Google Scholar 

  20. S. Echterhoff, On induced covariant systems, Proc. Am. Math. Soc., 108 (1990), 703–708.

    Google Scholar 

  21. S. Echterhoff, Morita equivalent actions and a new version of the Packer-Raeburn stabilization trick, J. Lond. Math. Soc., II. Ser., 50 (1994), 170–186.

  22. G. Elliott, T. Natsume and R. Nest, The Heisenberg group and K-theory, K-Theory, 7 (1993), 409–428.

    Google Scholar 

  23. J. Fell, The structure of algebras of operator fields, Acta Math., 106 (1961), 233–280.

    Google Scholar 

  24. J. Glimm, Locally compact transformation groups, Trans. Am. Math. Soc., 101 (1961), 124–138.

    Google Scholar 

  25. P. Green, The local structure of twisted covariance algebras, Acta. Math., 140 (1978), 191–250.

    Google Scholar 

  26. N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math., 144 (2001), 23–74.

    Google Scholar 

  27. G. P. Hochschild, Basic theory of algebraic groups and Lie algebras, Springer, GTM 75, 1981.

  28. R. Howe, The Fourier transform for nilpotent locally compact groups: I, Pac. J. Math., 73 (1977), 307–327.

    Google Scholar 

  29. G. Kasparov, Operator K-theory and its applications: Elliptic operators, group representations, higher signatures, C*-extensions, in: Proc. Internat. Congress of Mathematicians, vol. 2, Warsaw, 1983, 987–1000.

  30. G. Kasparov, The operator K-functor and extensions of C*-algebras, Math. USSR Izvestija 16, no. 3 (1981), 513–572.

    Google Scholar 

  31. G. Kasparov, K-theory, group C*-algebras, higher signatures (Conspectus), in: Novikov conjectures, index theorems and rigidity. Lond. Math. Soc., Lect. Note Ser., 226 (1995), 101–146.

  32. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91 (1988), 147–201.

    Google Scholar 

  33. G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, To appear in Ann. Math.

  34. E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C*-algebras, Math. Ann., 315 (1999), 169–203.

    Google Scholar 

  35. E. Kirchberg and S. Wassermann, Permanence properties of C*-exact groups, Doc. Math., 5 (2000), 513–558.

    Google Scholar 

  36. H. Kraft, P. Slodowy and T. A. Springer, Algebraische Transformationsgruppen und Invariantentheorie, DMV-Seminar, Band 13, Birkhäuser 1989.

  37. V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, PhD Dissertation, Universite Paris Sud, 1999.

  38. V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math., 149 (2002), 1–95.

    Google Scholar 

  39. V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, Prog. Math., 202 (2001), 31–46.

    Google Scholar 

  40. V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, in: Proc. Internat. Congress of Mathematicians, Vol. III, Beijing, 2002.

  41. R. Y. Lee, On the C*-algebras of operator fields, Indiana Univ. Math. J., 25 (1976), 303–314.

    Google Scholar 

  42. G. Lion and P. Perrin, Extension des Representations de groupe unipotents p-adiques, Calculs d’obstructions, Lect. Notes Math., 880 (1981), 337–356.

    Google Scholar 

  43. C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. Math., 82 (1965), 146–182.

    Google Scholar 

  44. G. Mackey, Borel structure in groups and their duals, Trans. Am. Math. Soc., 85 (1957), 134–165.

    Google Scholar 

  45. D. Montgomery and L. Zippin, Topological transformation groups, Interscience Tracts in Pure and Applied Mathematics, New York: Interscience Publishers, Inc. XI, 1955.

  46. H. Oyono-Oyono, Baum-Connes conjecture and extensions, J. Reine Angew. Math., 532 (2001), 133–149.

    Google Scholar 

  47. J. Packer and I. Raeburn, Twisted crossed products of C*-algebras, Math. Proc. Camb. Philos. Soc., 106 (1989), 293–311.

  48. G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London, 1979.

  49. L. Pukánszky, Characters of connected Lie groups, Mathematical surveys and Monographs, Vol. 71, American Mathematical Society, Rhode Island 1999.

  50. J. Rosenberg, Group C*-algebras and topological invariants, in: Operator algebras and group representations, Proc. Int. Conf., Neptun/Rom. 1980, Vol. II, Monogr. Stud. Math., 18 (1984), 95–115.

  51. M. Rosenlicht, A remark on quotient spaces, An. Acad. Bras. Ciênc., 35 (1963), 487–489.

    Google Scholar 

  52. J.L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-theory, 16, no. 2 (1999), 129–184.

  53. A. Valette, K-theory for the reduced C*-algebra of a semi-simple Lie group with real rank 1 and finite centre, Oxford Q. J. Math., 35 (1984), 341–359.

  54. A. Wassermann, Une demonstration de la conjecture of Connes-Kasparov pour les groupes de Lie lineaires connexes reductifs, C. R. Acad. Sci., Paris, Sér. I, Math., 304 (1987), 559–562.

  55. A. Weil, Basic number theory, Die Grundlehren der Mathematischen Wissenschaften, Band 144, Springer, New York-Berlin, 1974.

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Authors and Affiliations

  1. Bât. de mathématiques, Université Blaise Pascal, 63177, Aubière, France

    Jérôme Chabert

  2. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149, Münster, Germany

    Siegfried Echterhoff

  3. Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen, Denmark

    Ryszard Nest

Authors
  1. Jérôme Chabert
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  2. Siegfried Echterhoff
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  3. Ryszard Nest
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Correspondence to Jérôme Chabert, Siegfried Echterhoff or Ryszard Nest.

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Dedicated to the memory of Peter Slodowy

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Chabert, J., Echterhoff, S. & Nest, R. The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups. Publ. Math. 97, 239–278 (2003). https://doi.org/10.1007/s10240-003-0014-2

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  • DOI: https://doi.org/10.1007/s10240-003-0014-2

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