Electronic Research Announcements

Operator $K$ -theory for groups which act properly and isometrically on Hilbert space

Author(s): Nigel Higson; Gennadi Kasparov
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131-142.
MSC (1991): Primary 46L20
Posted: December 19, 1997
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Abstract: Let $G$

be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the $K$

-theory groups of the $C^{*}$ -algebras $C^{*}_{\max }(G)$ and $C^{*}_{ \smash{\text{red}}}(G)$ . Our result is in accordance with the Baum-Connes conjecture.

1.: M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford 19 (1968), 113-140. MR 37:3584
2.: P. Baum, A. Connes, and N. Higson, Classifying space for proper $G$ -actions and $K$ -theory of group $C^{*}$ -algebras, Contemp. Math. 167 (1994), 241-291. MR 96c:46070
3.: M. E. B. Bekka, P.-A. Cherix, and A. Valette, Proper affine isometric actions of amenable groups, Novikov conjectures, Index theorems and rigidity, vol. 2, S. Ferry, A. Ranicki and J. Rosenberg, editors, Cambridge University Press, Cambridge, 1995, pp. 1-4. MR 97e:43001
4.: B. Blackadar, $K$ -theory for operator algebras, MSRI Publication Series 5, Springer-Verlag, New York-Heidelberg-Berlin-Tokyo, 1986. MR 88g:46082
5.: A. Connes, An analogue of the Thom isomorphism for crossed products, Advances in Math. 39 (1981), 31-55. MR 82j:46084
6.: A. Connes and N. Higson, Déformations, morphismes asymptotiques et $K$ -théorie bivariante, C. R. Acad. Sci. Paris 311 Série 1 (1990), 101-106. MR 91m:46114
7.: P. Delorme, 1-cohomologie des représentations unitaries des groupes de Lie semi-simples et résolubles. Produits tensoriels continus et représentations, Bull. Soc. Math. France 105 (1977), 281-336. MR 58:28272
8.: S. Ferry, A. Ranicki and J. Rosenberg, A history and survey of the Novikov conjecture, Novikov conjectures, Index theorems and rigidity, vol. 1, S. Ferry, A. Ranicki and J. Rosenberg, editors, Cambridge University Press, Cambridge, 1995, pp. 7-66. MR 97f:57036
9.: P. Green, Equivariant $K$ -theory and crossed product $C^{*}$ -algebras, Proceedings of Symposia in Pure Mathematics 38 Part 1 (1982), 337-338. MR 83j:46004a
10.: M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, G. A. Niblo and M. A. Roller, editors, Cambridge Univeristy Press, Cambridge, 1993, pp. 1-295. MR 95m:20041
11.: E. Guentner, N. Higson, and J. Trout, Equivariant $E$ -theory, Preprint, 1997.
12.: P. de la Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. de France, 1989. MR 90m:22001
13.: N. Higson and G. Kasparov, A note on the Baum-Connes conjecture in -theory and $E$ -theory, In preparation.
14.: N. Higson, G. Kasparov, and J. Trout, A Bott periodicity theorem for infinite-dimensional Euclidean space, Advances in Math. (to appear).
15.: P. Julg, $K$ -théorie equivariante et produits croisés, C. R. Acad. Sci. Paris 292 Série 1 (1981), 629-632. MR 83b:46090
16.: G. G. Kasparov, Equivariant -theory and the Novikov conjecture, Inventiones Math. 91 (1988), 147-201. MR 88j:58123
17.: E. Kirchberg and S. Wassermann, In preparation.
18.: J. Mingo and W. Phillips, Equivariant triviality theorems for Hilbert modules, Proc. Amer. Math. Soc. 91 (1984), 225-230. MR 85f:46111
19.: G. K. Pedersen, $C^{*}$ -algebras and their automorphism groups, Academic Press, London-New York-San Francisco, 1979. MR 81e:46037
20.: G. Segal, Equivariant $K$ -theory, Publ. Math. IHES 34 (1968), 129-151. MR 38:2769
21.: J.-L. Tu, The Baum-Connes conjecture and discrete group actions on trees, Preprint.
22.: S. Wassermann, Exact $C^{*}$ -algebras and related topics, Res. Inst. of Math. Lecture Note Series 19, Seoul National University, Seoul, South Korea, 1994. MR 95b:46081

Nigel Higson
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, PA 16802
Email: higson@math.psu.edu

Gennadi Kasparov
Affiliation: Institut de Mathématiques de Luminy, CNRS-Luminy-Case 930, 163 Avenue de Luminy 13288, Marseille Cedex 9, France
Email: kasparov@iml.univ-mrs.fr

DOI: 10.1090/S1079-6762-97-00038-3
PII: S 1079-6762(97)00038-3
Keywords: Baum-Connes conjecture, $C^{*}$-algebras, $K$-theory
Received by editor(s): October 25, 1997
Posted: December 19, 1997
Additional Notes: The first author was partially supported by an NSF grant.
Communicated by: Masamichi Takesaki
Copyright of article: Copyright 1997, American Mathematical Society

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