The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups
Publications Mathématiques de l'IHÉS, Volume 97 (2003), p. 239-278

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.

@article{PMIHES_2003__97__239_0,
     author = {Chabert, J\'er\^ome and Echterhoff, Siegfried and Nest, Ryszard},
     title = {The Connes-Kasparov conjecture for almost connected groups and for linear $p$-adic groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {97},
     year = {2003},
     pages = {239-278},
     doi = {10.1007/s10240-003-0014-2},
     zbl = {1048.46057},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2003__97__239_0}
}
Chabert, Jérôme; Echterhoff, Siegfried; Nest, Ryszard. The Connes-Kasparov conjecture for almost connected groups and for linear $p$-adic groups. Publications Mathématiques de l'IHÉS, Volume 97 (2003) pp. 239-278. doi : 10.1007/s10240-003-0014-2. http://www.numdam.org/item/PMIHES_2003__97__239_0/

1. H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann., 212 (1974), 1-19. | MR 375264 | Zbl 0276.57019

2. M. Atiyah, R. Bott and A. Shapiro, Clifford Modules, Topology 3, Suppl. 1 (1964), 3-38. | MR 167985 | Zbl 0146.19001

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. | MR 1292018 | Zbl 0830.46061

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. | MR 1467072 | Zbl 0918.46061

5. B. Blackadar, K-theory for operator algebras, MSRI Pub. 5, Springer 1986. | MR 859867 | Zbl 0597.46072

6. E. Blanchard, Deformations de C*-algebres de Hopf, Bull. Soc. Math. Fr., 124 (1996), 141-215. | Numdam | MR 1395009 | Zbl 0851.46040

7. J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer 1998. | MR 1659509 | Zbl 0912.14023

8. A. Borel, Linear Algebraic Groups, Springer, GTM 126 (1991). | MR 1102012 | Zbl 0726.20030

9. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535. | MR 147566 | Zbl 0107.14804

10. A. Borel and J. Tits, Groupes reductifs, Inst. Hautes Études Sci., Publ. Math., 27 (1965), 55-150. | Numdam | MR 207712 | Zbl 0145.17402

11. J. Chabert and S. Echterhoff, Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions, K-Theory, 23 (2001), 157-200. | MR 1857079 | Zbl 1010.19004

12. J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6 (2001), 127-183. | MR 1836047 | Zbl 0984.46047

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. | MR 1841893 | Zbl 1003.46037

14. J. Chabert, S. Echterhoff and H. Oyono-Oyono, Going-Down functors, the Künneth formula, and the Baum-Connes conjecture, Preprintreihe SFB 478, Münster. | MR 2100669 | Zbl 1063.46056

15. P.-A. Cherix, M. Cowling, P. Jolisssaint, P. Julg and A. Valette, Groups with the Haagerup property, Progress in Mathematics 197, Birkhäuser 2000. | MR 1852148 | Zbl 1030.43002

16. C. Chevalley, Theorie des groupes de Lie, Groupes algebriques, Theoremes generaux sur les algebres de Lie, 2ieme ed., Hermann & Cie. IX, Paris, 1968. | Zbl 0186.33104

17. A. Connes and H. Moscovici, The L2-index theorem for homogeneous spaces of Lie groups, Ann. Math., 115 (1982), 291-330. | MR 647808 | Zbl 0515.58031

18. J. Dixmier, C*- algebras (English Edition). North Holland Publishing Company 1977. | Zbl 0372.46058

19. M. Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math., 149 (1983), 153-213. | MR 688348 | Zbl 0529.22011

20. S. Echterhoff, On induced covariant systems, Proc. Am. Math. Soc., 108 (1990), 703-708. | MR 994776 | Zbl 0692.46054

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. | MR 1277761 | Zbl 0807.46081

22. G. Elliott, T. Natsume and R. Nest, The Heisenberg group and K-theory, K-Theory, 7 (1993), 409-428. | MR 1255059 | Zbl 0803.46076

23. J. Fell, The structure of algebras of operator fields, Acta Math., 106 (1961), 233-280. | MR 164248 | Zbl 0101.09301

24. J. Glimm, Locally compact transformation groups, Trans. Am. Math. Soc., 101 (1961), 124-138. | MR 136681 | Zbl 0119.10802

25. P. Green, The local structure of twisted covariance algebras, Acta. Math., 140 (1978), 191-250. | MR 493349 | Zbl 0407.46053

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. | MR 1821144 | Zbl 0988.19003

27. G. P. Hochschild, Basic theory of algebraic groups and Lie algebras, Springer, GTM 75, 1981. | MR 620024 | Zbl 0589.20025

28. R. Howe, The Fourier transform for nilpotent locally compact groups: I, Pac. J. Math., 73 (1977), 307-327. | MR 492059 | Zbl 0396.43013

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. | MR 804752 | Zbl 0571.46047

30. G. Kasparov, The operator K-functor and extensions of C*-algebras, Math. USSR Izvestija 16, no. 3 (1981), 513-572. | MR 582160 | Zbl 0464.46054

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. | MR 1388299 | Zbl 0957.58020

32. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91 (1988), 147-201. | MR 918241 | Zbl 0647.46053

33. G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, To appear in Ann. Math. | Zbl 1029.19003

34. E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C*-algebras, Math. Ann., 315 (1999), 169-203. | MR 1721796 | Zbl 0946.46054

35. E. Kirchberg and S. Wassermann, Permanence properties of C*-exact groups, Doc. Math., 5 (2000), 513-558. | MR 1725812 | Zbl 0958.46036

36. H. Kraft, P. Slodowy and T. A. Springer, Algebraische Transformationsgruppen und Invariantentheorie, DMV-Seminar, Band 13, Birkhäuser 1989. | MR 1044582 | Zbl 0682.00008

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. | MR 1914617 | Zbl 1084.19003

39. V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, Prog. Math., 202 (2001), 31-46. | MR 1905349 | Zbl 1030.19002

40. V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, in: Proc. Internat. Congress of Mathematicians, Vol. III, Beijing, 2002. | MR 1957086 | Zbl 0997.19003

41. R. Y. Lee, On the C*-algebras of operator fields, Indiana Univ. Math. J., 25 (1976), 303-314. | MR 410400 | Zbl 0322.46062

42. G. Lion and P. Perrin, Extension des Representations de groupe unipotents p-adiques, Calculs d'obstructions, Lect. Notes Math., 880 (1981), 337-356. | Zbl 0463.22014

43. C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. Math., 82 (1965), 146-182. | MR 181701 | Zbl 0139.30702

44. G. Mackey, Borel structure in groups and their duals, Trans. Am. Math. Soc., 85 (1957), 134-165. | MR 89999 | Zbl 0082.11201

45. D. Montgomery and L. Zippin, Topological transformation groups, Interscience Tracts in Pure and Applied Mathematics, New York: Interscience Publishers, Inc. XI, 1955. | MR 73104 | Zbl 0068.01904

46. H. Oyono-Oyono, Baum-Connes conjecture and extensions, J. Reine Angew. Math., 532 (2001), 133-149. | MR 1817505 | Zbl 0973.46064

47. J. Packer and I. Raeburn, Twisted crossed products of C*-algebras, Math. Proc. Camb. Philos. Soc., 106 (1989), 293-311. | MR 1002543 | Zbl 0757.46056

48. G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London, 1979. | MR 548006 | Zbl 0416.46043

49. L. Pukánszky, Characters of connected Lie groups, Mathematical surveys and Monographs, Vol. 71, American Mathematical Society, Rhode Island 1999. | MR 1707323 | Zbl 0934.22002

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. | MR 733308 | Zbl 0524.22009

51. M. Rosenlicht, A remark on quotient spaces, An. Acad. Bras. Ciênc., 35 (1963), 487-489. | MR 171782 | Zbl 0123.13804

52. J.L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-theory, 16, no. 2 (1999), 129-184. | MR 1671260 | Zbl 0932.19005

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. | MR 755672 | Zbl 0545.22006

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. | MR 894996 | Zbl 0615.22011

55. A. Weil, Basic number theory, Die Grundlehren der Mathematischen Wissenschaften, Band 144, Springer, New York-Berlin, 1974. | MR 234930 | Zbl 0326.12001