Abstract
In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.
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References
[BG]R. Beals, P. Greiner, Calculus on Heisenberg manifolds, Annals of Math. Studies 119, Princeton Univ. Press, Princeton, N. J., 1988.
[C1]A. Connes, Noncommutative Geometry, Academic Press, Inc. 1994.
[C2]A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, Geometric methods in operator algebras, (Kyoto, 1983), pp. 52–144, Pitman Res. Notes in Math. 123 Longman, Harlow (1986).
[C3]A. Connes, Noncommutative Geometry and Physics, Les Houches Conference,
[CL]A. Connes, J. Lott, Particle models and noncommutative geometry, Nuclear Physics B 18B (1990), suppl. 29–47 (1991).
[CM1]A. Connes, H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345–388.
[CM2]A. Connes, H. Moscovici, Transgression and the Chern character of finitedimensionalK-cycles, Commun. Math. Phys. 155 (1993), 103–122.
[G]P. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Math. Lecture Ser. 11, Publish or Perish, Wilmington, Del., 1984.
[HSk]M. Hilsum, G. Skandalis, MorphismesK-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov, Ann. Sci. École Norm. Sup. (4) 20 (1987), 325–390.
[KV]M. Kontsevich, S. Vishik, Determinants of elliptic pseudodifferential operators, Preprint, 1994.
[S]R. T. Seeley, Complex powers of elliptic operators, Proc. Symp. Pure Math. 10 (1967), 288–307.
[W]M. Wodzicki, Noncommutative residue, Part I. Fundamentals,K-theory, arithmetic and geometry (Moscow, 1984–86), 320–399; Springer Lecture Notes Math. 1289, Berlin, (1987).
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We dedicate this paper to Misha Gromov
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Connes, A., Moscovici, H. The local index formula in noncommutative geometry. Geometric and Functional Analysis 5, 174–243 (1995). https://doi.org/10.1007/BF01895667
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DOI: https://doi.org/10.1007/BF01895667