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Closed star products and cyclic cohomology. (English) Zbl 0767.55005

From the authors’ abstract: We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold \(W\)) is closed iff integration over \(W\) is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in the usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well- defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.
Reviewer: P.J.Kahn (Ithaca)

MSC:

55N35 Other homology theories in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19K56 Index theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
58J40 Pseudodifferential and Fourier integral operators on manifolds
57R20 Characteristic classes and numbers in differential topology
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[1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Ann. Physics 111, 61-151 (1978). · Zbl 0377.53024
[2] Connes, A., Géométrie non commutative, Interéditions, Paris (1990); Publ. Math. IHES 62, 41-144 (1985). · Zbl 0592.46056
[3] Connes, A. and Higson, N., C.R. Acad. Sci. Paris Sér I Math. 311, 101-106 (1990).
[4] Agarwal, G. S. and Wolf, E., Phys. Rev. D 2, 2161 (1970). · Zbl 1227.81196
[5] Lichnerowicz, A., Ann. Inst. Fourier 32, 157-209 (1982).
[6] Lecomte, P. and De Wilde, M., Note Mat (Koethe special issue; Vol. X, in press) and references quoted therein; Omori, H., Maeda, Y., and Yoshioka, A., Adv. in Math. 85, 224-255 (1991).
[7] Widom, H., Bull. Sci. Math. (2)104, 19-63 (1980); See also Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York (1978), pp. 345-395.
[8] Connes, A. and Moscovici, H., Topology 20, 345-388 (1990); Connes, A., Gromov, M., and Moscovici, H., C.R. Acad. Sci. Paris Sér I Math. 310, 273-277 (1990). · Zbl 0759.58047
[9] Gilkey, P. B., Invariance Theory, the Heat Equation, and the Atiyah-Singer Theorem, Publish or Perish, Wilmington, Del., 1984. · Zbl 0565.58035
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