×

zbMATH — the first resource for mathematics

The Connes-Kasparov conjecture for almost connected groups and for liner \(p\)-adic groups. (English) Zbl 1048.46057
Let \(G\) be a second countable locally compact group and let \(G_0\) be the connected component of the neutral element. It is proved that if \(G/G_0\) satisfies the Baum-Connes conjecture for arbitrary coefficients then \(G\) satisfies this conjecture for coefficients being the \(C^*\)-algebra of compact operators with an arbitrary action of \(G\). This implies that if \(G\) is a second countable almost connected group then it satisfies the Baum-Connes conjecture with trivial coefficients. The last conclusion is proved also for \(G\) being in the group of \(k\)-rational points of a linear algebraic group over a local field \(k\) of characteristic zero. As a corollary of the main result, it is shown that all square-integrable factor representations of a connected unimodular Lie group \(G\) are of type I and that \(G\) has no square-integrable factor representations if \(\dim(G/L)\) is odd, where \(L\subset G\) denotes the maximal compact subgroup.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
20G25 Linear algebraic groups over local fields and their integers
19K35 Kasparov theory (\(KK\)-theory)
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
PDF BibTeX XML Cite
Full Text: DOI arXiv Numdam EuDML
References:
[1] H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann., 212 (1974), 1–19. · Zbl 0276.57019
[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. · 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. · Zbl 0918.46061
[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.
[7] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer 1998. · Zbl 0912.14023
[8] A. Borel, Linear Algebraic Groups, Springer, GTM 126 (1991). · Zbl 0726.20030
[9] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485–535. · Zbl 0107.14804
[10] A. Borel and J. Tits, Groupes reductifs, Inst. Hautes Études Sci., Publ. Math., 27 (1965), 55–150. · 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. · Zbl 1010.19004
[12] J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6 (2001), 127–183. · 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. · 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. · 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.
[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. · Zbl 0515.58031
[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.
[20] S. Echterhoff, On induced covariant systems, Proc. Am. Math. Soc., 108 (1990), 703–708. · 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. · Zbl 0807.46081
[22] G. Elliott, T. Natsume and R. Nest, The Heisenberg group and K-theory, K-Theory, 7 (1993), 409–428. · Zbl 0803.46076
[23] J. Fell, The structure of algebras of operator fields, Acta Math., 106 (1961), 233–280. · Zbl 0101.09301
[24] J. Glimm, Locally compact transformation groups, Trans. Am. Math. Soc., 101 (1961), 124–138. · Zbl 0119.10802
[25] P. Green, The local structure of twisted covariance algebras, Acta. Math., 140 (1978), 191–250. · 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. · Zbl 0988.19003
[27] G. P. Hochschild, Basic theory of algebraic groups and Lie algebras, Springer, GTM 75, 1981. · Zbl 0589.20025
[28] R. Howe, The Fourier transform for nilpotent locally compact groups: I, Pac. J. Math., 73 (1977), 307–327. · 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.
[30] G. Kasparov, The operator K-functor and extensions of C*-algebras, Math. USSR Izvestija 16, no. 3 (1981), 513–572. · 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. · Zbl 0957.58020
[32] G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math., 91 (1988), 147–201. · 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. · Zbl 0946.46054
[35] E. Kirchberg and S. Wassermann, Permanence properties of C*-exact groups, Doc. Math., 5 (2000), 513–558. · Zbl 0958.46036
[36] H. Kraft, P. Slodowy and T. A. Springer, Algebraische Transformationsgruppen und Invariantentheorie, DMV-Seminar, Band 13, Birkhäuser 1989. · 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. · Zbl 1084.19003
[39] V. Lafforgue, Banach KK-Theory and the Baum-Connes conjecture, Prog. Math., 202 (2001), 31–46.
[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. · 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. · Zbl 0139.30702
[44] G. Mackey, Borel structure in groups and their duals, Trans. Am. Math. Soc., 85 (1957), 134–165. · 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. · Zbl 0068.01904
[46] H. Oyono-Oyono, Baum-Connes conjecture and extensions, J. Reine Angew. Math., 532 (2001), 133–149. · Zbl 0973.46064
[47] J. Packer and I. Raeburn, Twisted crossed products of C*-algebras, Math. Proc. Camb. Philos. Soc., 106 (1989), 293–311. · Zbl 0757.46056
[48] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London, 1979. · Zbl 0416.46043
[49] L. Pukánszky, Characters of connected Lie groups, Mathematical surveys and Monographs, Vol. 71, American Mathematical Society, Rhode Island 1999. · 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.
[51] M. Rosenlicht, A remark on quotient spaces, An. Acad. Bras. Ciênc., 35 (1963), 487–489. · Zbl 0123.13804
[52] J.L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-theory, 16, no. 2 (1999), 129–184. · 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. · 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. · Zbl 0615.22011
[55] A. Weil, Basic number theory, Die Grundlehren der Mathematischen Wissenschaften, Band 144, Springer, New York-Berlin, 1974.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.