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Induction for Banach Algebras, Groupoids and KKban

Published online by Cambridge University Press:  23 October 2009

Walther Paravicini
Walther Paravicini
Affiliation:
Mathematisches Institut der WWU Münster, Einsteinstr. 62, 48149 Münster, Germany, W.Paravicini@uni-muenster.de
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Abstract

Given two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).

Keywords

Locally compact groupoidKKban-theoryBanach algebraBaum-Connes conjectureBost conjecture
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 3 , December 2009 , pp. 405 - 468
Copyright
Copyright © ISOPP 2009

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References

BEL07.Bartels, Arthur, Echterhoff, Siegfried, and Lück, Wolfgang. Inheritance of isomorphism conjectures under colimits. In K-Theory and Noncommutative Geometry (Valladolid 2006), Editors: Cortinas, Guillermo, Cuntz, Joachim, Karoubi, Max, Nest, Ryszard, Weibel, Charles A.. Series of Congress Reports of the European Mathematical Society 2008, pages 41–70.Google Scholar
Bla96.Blanchard, Étienne. Déformations de C*-algèbres de Hopf. Bull. Soc. Math. France, 124(1):141–215, 1996.CrossRefGoogle Scholar
Bos90.Bost, Jean-Benoit. Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs. Invent. Math., 101(2):261–333, 1990.CrossRefGoogle Scholar
Con82.Connes, A.. A survey of foliations and operator algebras. In Operator algebras and applications, Part I (Kingston, Ont., 1980), volume 38 of Proc. Sympos. Pure Math., pages 521–628. Amer. Math. Soc., Providence, R.I., 1982.CrossRefGoogle Scholar
DL98.Davis, James F. and Lück, Wolfgang. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory, 15(3):201–252, 1998.CrossRefGoogle Scholar
Hae84.Haefliger, André. Groupoïdes d'holonomie et classifiants. Astérisque, (116):70–97, 1984. Transversal structure of foliations (Toulouse, 1982).Google Scholar
HS87.Hilsum, Michel and Skandalis, Georges. Morphismes K-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes). Ann. Sci. École Norm. Sup. (4), 20(3):325–390, 1987.CrossRefGoogle Scholar
Kas80.Kasparov, G. G.. The operator K-functor and extensions of C*-algebras. Izv. Akad. Nauk SSSR Ser. Mat., 44(3):571–636, 719, 1980.Google Scholar
Kas88.Kasparov, Gennadi G.. Equivariant KK-theory and the Novikov conjecture. Invent. Math., 91:147–201, 1988.CrossRefGoogle Scholar
Laf02.Lafforgue, Vincent. K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math., 149:1–95, 2002.CrossRefGoogle Scholar
Laf06.Lafforgue, Vincent. K-théorie bivariante pour les algèbres de Banach, groupoïdes et conjecture de Baum-Connes. Avec un appendice d'Hervé Oyono-Oyono. J. Inst. Math. Jussieu, 2006. Published online by Cambridge University Press 28 Nov 2006.Google Scholar
LG94.Le Gall, Pierre-Yves. Théorie de Kasparov équivariante et groupoïdes. PhD thesis, Université Paris VII, 1994.Google Scholar
LG99.Le Gall, Pierre-Yves. Théorie de Kasparov équivariante et groupoïdes. I. K-Theory, 16(4):361–390, 1999.CrossRefGoogle Scholar
MRW87.Muhly, Paul S., Renault, Jean N., and Williams, Dana P.. Equivalence and isomorphism for groupoid C*-algebras. J. Operator Theory, 17(1):3–22, 1987.Google Scholar
Par07a.Paravicini, Walther. A generalised Green-Julg theorem for proper groupoids and Banach algebras. Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik, 480, 2007.Google Scholar
Par07b.Paravicini, Walther. KK-Theory for Banach Algebras And Proper Groupoids. PhD thesis, Universität Münster, 2007. Persistent identifier: urn:nbn:de:hbz:6-39599660289.Google Scholar
Par08.Paravicini, Walther. Morita equivalences and KK-theory for Banach algebras. J. of the Inst. of Math. of Jussieu, Forthcoming(-1):1–29, 2008.Google Scholar
Ren80.Renault, Jean N.. A Groupoid Approach to C*-Algebras, volume 793. Springer-Verlag, Berlin, 1980. Lecture Notes in Mathematics.CrossRefGoogle Scholar
Ren82.Renault, Jean. C*-algebras of groupoids and foliations. In Kadison, Richard V., editor, Proceedings of the Symposium in Pure Mathematics of the AMS, held at Queens University, Kingston, Ontario, 1980, pages 339–350, 1982.Google Scholar
Rie74.Rieffel, Marc A.. Induced representations of C*-algebras. Advances in Math., 13:176–257, 1974.CrossRefGoogle Scholar
Rie76.Rieffel, Marc A.. Strong Morita equivalence of certain transformation group C*- algebras. Math. Ann., 222(1):7–22, 1976.CrossRefGoogle Scholar
Tu99.Tu, Jean-Louis. La conjecture de Novikov pour les feuilletages hyperboliques. K-Theory, 16(2):129–184, 1999.CrossRefGoogle Scholar
Tu00.Tu, Jean-Louis The Baum-Connes conjecture for groupoids. In C*-algebras (Münster, 1999), pages 227–242. Springer, Berlin, 2000.CrossRefGoogle Scholar
Tu04.Tu, Jean-Louis. Non-Hausdorff groupoids, proper actions and K-theory. Doc. Math., 9:565–597 (electronic), 2004.CrossRefGoogle Scholar