Operator $K$-theory for groups which act properly and isometrically on Hilbert space
Authors:
Nigel Higson and Gennadi Kasparov
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 131-142
MSC (1991):
Primary 46L20
DOI:
https://doi.org/10.1090/S1079-6762-97-00038-3
Published electronically:
December 19, 1997
MathSciNet review:
1487204
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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.
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- 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.
- 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.
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- E. Guentner, N. Higson, and J. Trout, Equivariant $E$-theory, Preprint, 1997.
- P. de la Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. de France, 1989.
- N. Higson and G. Kasparov, A note on the Baum-Connes conjecture in $KK$-theory and $E$-theory, In preparation.
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- J. Mingo and W. Phillips, Equivariant triviality theorems for Hilbert modules, Proc. Amer. Math. Soc. 91 (1984), 225–230.
- G. K. Pedersen, $C^{*}$-algebras and their automorphism groups, Academic Press, London–New York–San Francisco, 1979.
- G. Segal, Equivariant $K$-theory, Publ. Math. IHES 34 (1968), 129–151.
- J.-L. Tu, The Baum-Connes conjecture and discrete group actions on trees, Preprint.
- S. Wassermann, Exact $C^{*}$-algebras and related topics, Res. Inst. of Math. Lecture Note Series 19, Seoul National University, Seoul, South Korea, 1994.
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Additional Information
Nigel Higson
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
MR Author ID:
238781
ORCID:
0000-0001-9661-1663
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
Keywords:
Baum-Connes conjecture,
$C^{*}$-algebras,
$K$-theory
Received by editor(s):
October 25, 1997
Published electronically:
December 19, 1997
Additional Notes:
The first author was partially supported by an NSF grant.
Communicated by:
Masamichi Takesaki
Article copyright:
© Copyright 1997
American Mathematical Society