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The analytic index of elliptic pseudodifferential operators on a singular foliation

Published online by Cambridge University Press:  10 March 2011

Iakovos Androulidakis  and
Georges Skandalis
Iakovos Androulidakis
Affiliation:
Georg-August Universität Göttingen, Mathematisches Institut, Bunsenstrasse 3-5, D-37073 Göttingen, Germany, [email protected]
Georges Skandalis
Affiliation:
Université Paris Diderot (Paris 7) - CNRS, Institut de Mathématiques de Jussieu, UMR 7586 175, rue du Chevaleret, F-75013 Paris, France, [email protected]
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Abstract

In previous papers ([1, 2]) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,). In the current paper we construct the analytic index of an elliptic operator as a KK-theory element, and prove that this element can be obtained from an “adiabatic foliation” on M×ℝ, which we introduce here.

Keywords

analytic indexfoliationsdeformationC*-algebra.
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 3 , December 2011 , pp. 363 - 385
Copyright
Copyright © ISOPP 2011

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References

1.Androulidakis, I. and Skandalis, G., The holonomy groupoid of a singular foliation. J. Reine Angew. Math. 626(2009) 1–37.CrossRefGoogle Scholar
2 .Androulidakis, I. and Skandalis, G., Pseuodifferential calculus on a singular foliation. arXiv:math.DG/0909.1342Google Scholar
3.Atiyah, M.F. and Singer, I.M., The index of elliptic operators I. Ann. of Math. 87(1968) 484–530.CrossRefGoogle Scholar
4.Atiyah, M.F. and Singer, I.M., The index of elliptic operators IV. Ann. of Math. 93(1971) 119–138.CrossRefGoogle Scholar
5.Bigonnet, B., Pradines, J., Graphe d'un feuilletage singulier, C. R. Acad. Sci. Paris 300(13) (1985) 439–442Google Scholar
6.Rouse, Paulo Carrillo, An analytic index for Lie groupoids. In K-theory and noncommutative geometry, EMS Ser. Congr. Rep. 2008 (Eur. Math. Soc., Zürich), pp. 181–199. (Preprint arxiv:math.KT/0612455)Google Scholar
7.Connes, A., Sur la théorie non commutative de l'intégration. in Algèbres d'opérateurs Springer Lect. Notes in Math. 725 (1979) 19–143.CrossRefGoogle Scholar
8.Connes, A., Noncommutative Geometry. Academic Press, 1984.Google Scholar
9.Connes, A. and Skandalis, G., The Longitudinal Index Theorem for Foliations. Pub. of Research Institute for Math. Sc. 20, No 6 (1984) 1139–1183.CrossRefGoogle Scholar
10.Cuntz, J. and Skandalis, G., Mapping cones and exact sequences in KK-theory J. Operator theory 15 (1986) 163–180.Google Scholar
11.Kasparov, G.G., The operator K-functor and extensions of C*-algebras. Math. USSR Izv. 16 (1981) n° 3, 513–572. Translated from Izv. Akad. Nauk. S.S.S.R. Ser. Mat. 44 (1980) 571–636.CrossRefGoogle Scholar
12.Monthubert, B. and Pierrot, F., Indice analytique et groupoïdes de Lie. C. R. Acad. Sci. Paris Ser. I, 325 (1997) no. 2, 193–198.CrossRefGoogle Scholar
13.Nistor, V., Weinstein, A. and Xu, P., Pseudodifferential operators on differential groupoids. Pacific J. Math. 189 (1999) no. 1, 117–152.CrossRefGoogle Scholar
14.Renault, J.N., A groupoid approach to C*-algebras. L.N. in Math. 793 (1979).Google Scholar
15.Stefan, P., Accessible sets, orbits, and foliations with singularities. Proc. London Math. Soc. (3) 29 (1974) 699–713.CrossRefGoogle Scholar
16.Sussmann, H. J., Orbits of families of vector fields and integrability of distributions. Trans. of the A. M. S. 180 (1973) 171–188CrossRefGoogle Scholar
17.Vassout, S., Unbounded pseudodifferential Calculus on Lie groupoids, Journal of Functional Analysis 236 (2006) 161–200CrossRefGoogle Scholar
18.Winkelnkemper, H. E., The graph of a foliation. Ann. Glob. Analysis and Geometry 1 (3) (1983) 51–75.CrossRefGoogle Scholar