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The longitudinal index theorem for foliations. (English) Zbl 0575.58030
A topological formula for the analytical index of a differential operator D elliptic along the leaves of the foliation (V,F) is given. The index in the Atiyah-Singer theorem for families is an element of the \(K^ 0(B)=K_ 0(C(B))\), where B is the basic space of the fibration. On the case of foliations the algebra C(B) is replaced by a canonically defined \(C^*\)-algebra \(C^*(V;F)\) and \(ind(D)\in K_ 0(C^*(V;F))\). The Kasparov K-bifunctor is a basic tool to prove the results of the paper.
Reviewer: V.Deundjak

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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[1] M.F. Atiyah and I. Singer, The index of elliptic operators I, Ann. of Math., 87 (1968), 484-530. · Zbl 0164.24001
[2] - , The index of elliptic operators IV, Ami. of Math., 93 (1971), 119-1 38.
[3] M.F. Atiyah, R. Bott and Shapiro, Clifford Modules, Topology, 3, suppl. 1 (1964), 3-38. · Zbl 0146.19001
[4] P. Baum, Cycles, Cocycles and K theory, to appear.
[5] P. Baum and A. Connes, Geometric ^-theory for Lie groups and Foliations, preprint. [6] P. Baum and R. Douglas, K-holomology and index theory, Proceedings of A.M.S., 38.
[6] L. Boutet de Montvel, A course on pseudodifferential operators and their applications, Duke Univ. Math. Series, II. · Zbl 0348.35002
[7] A. Connes, An analogue of the Thorn isomorphism, for crossed products of a C*- algebra by an action of R, Adv. in Math., 39 (1981). · Zbl 0461.46043
[8] , Sur la theorie non commutative de 1’integration, Lecture Notes in Math., 725, (1979), Springer, 19-143.
[9] [11] A. Connes et G. Skandalis. Theoreme de 1’indice pour les feuilletages, C.R. Acad. Sci. Paris, 292 (1981), 871-876. · Zbl 0529.58030
[10] J. Dixmier, Les C*-algebres et lews representations 2eme edition, Paris Gauthier Villars, 1969.
[11] 183-194.
[12] J. Hayden and RJ. Plymen, On the invariant of Dixmier-Douady^r^nw/1.
[13] M.W. Hirsh, Differential topology, Graduate texts in Math. 33, Springer Verlag, Berlin.
[14] L. Hormander, On the index of pseudodifferential operators, Elliptische Differential- gleischungen Band II, Koll., Berlin, 1969.
[15] G.G. Kasparov, Topological invariants of elliptic operators. I: /Chomology, Math. USSR Izv., 9 (1975), 751-792. · Zbl 0337.58006
[16] , Hilbert C*-modules: Theorems of Stinespring and Voiculescu, Journal of operator theory, 4 (1980). · Zbl 0456.46059
[17] , Operator K functor and extensions of C*-algebras, Izv. Akad. Nauk, S.S.S.R. ser. Mat., 44 (1980), 571-636.
[18] , ^-theory, group C*-algebras and higher signatures (Conspectus), part 1-2, preprint. · Zbl 0957.58020
[19] J.M. Kister, Microbundles are bundles, Ann. of Math., 80 (1964), 190-199. · Zbl 0131.20602
[20] B. Lawson, Foliations, Bull. Am. Math. Soc., 80 (1974), 369-418. · Zbl 0293.57014
[21] J. Milnor, Microbundles I, Topology, 3, Suppl. 1 (1964), 53.
[22] R.S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Math. Studies, 57 (1965). · Zbl 0202.23103
[23] I. Singer, Future extensions of index theory and elliptic operators, Prospects in Math., Annals, of Math. Studies, 70. · Zbl 0286.58011
[24] E. Winkelnkemper, The graph of a foliation, preprint. · Zbl 0526.53039
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