×

Operator \(K\)-theory for groups which act properly and isometrically on Hilbert space. (English) Zbl 0888.46046

Summary: 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.

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)

References:

[1] M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. (2) 19 (1968), 113 – 140. · Zbl 0159.53501 · doi:10.1093/qmath/19.1.113
[2] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and \?-theory of group \?*-algebras, \?*-algebras: 1943 – 1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240 – 291. · Zbl 0830.46061 · doi:10.1090/conm/167/1292018
[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 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 1 – 4. · Zbl 0959.43001 · doi:10.1017/CBO9780511629365.003
[4] Bruce Blackadar, \?-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. · Zbl 0597.46072
[5] A. Connes, An analogue of the Thom isomorphism for crossed products of a \?*-algebra by an action of \?, Adv. in Math. 39 (1981), no. 1, 31 – 55. · Zbl 0461.46043 · doi:10.1016/0001-8708(81)90056-6
[6] Alain Connes and Nigel Higson, Déformations, morphismes asymptotiques et \?-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101 – 106 (French, with English summary). · Zbl 0717.46062
[7] Patrick Delorme, 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations, Bull. Soc. Math. France 105 (1977), no. 3, 281 – 336 (French). · Zbl 0404.22006
[8] Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg, A history and survey of the Novikov conjecture, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 7 – 66. · Zbl 0954.57018 · doi:10.1017/CBO9780511662676.003
[9] Operator algebras and applications. Part 1, Proceedings of Symposia in Pure Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1982. Edited by Richard V. Kadison. Richard V. Kadison , Operator algebras and applications. Part 2, Proceedings of Symposia in Pure Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1982.
[10] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1 – 295. · Zbl 0841.20039
[11] E. Guentner, N. Higson, and J. Trout, Equivariant \(E\)-theory, Preprint, 1997.
[12] Pierre de la Harpe and Alain Valette, La propriété (\?) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989), 158 (French, with English summary). With an appendix by M. Burger. · Zbl 0759.22001
[13] N. Higson and G. Kasparov, A note on the Baum-Connes conjecture in \(KK\)-theory and \(E\)-theory, In preparation. · Zbl 0988.19003
[14] N. Higson, G. Kasparov, and J. Trout, A Bott periodicity theorem for infinite-dimensional Euclidean space, Advances in Math. (to appear). · Zbl 0911.46040
[15] Pierre Julg, \?-théorie équivariante et produits croisés, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 13, 629 – 632 (French, with English summary). · Zbl 0461.46044
[16] G. G. Kasparov, Equivariant \?\?-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147 – 201. · Zbl 0647.46053 · doi:10.1007/BF01404917
[17] E. Kirchberg and S. Wassermann, In preparation.
[18] J. A. Mingo and W. J. Phillips, Equivariant triviality theorems for Hilbert \?*-modules, Proc. Amer. Math. Soc. 91 (1984), no. 2, 225 – 230. · Zbl 0546.46049
[19] Gert K. Pedersen, \?*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. · Zbl 0416.46043
[20] Graeme Segal, Equivariant \?-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129 – 151. · Zbl 0199.26202
[21] J.-L. Tu, The Baum-Connes conjecture and discrete group actions on trees, Preprint. · Zbl 0939.19002
[22] Simon Wassermann, Exact \?*-algebras and related topics, Lecture Notes Series, vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. · Zbl 0828.46054
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.