Star products and local line bundles
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1581-1600

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([3]) see also Fedosov’s construction in ([7]). This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki ([18]) and Guillemin ([10]).

Nous définissons les fibrés en droites locaux sur une variété, qui sont classifiés par la cohomologie réelle de degré 2. Le twist d’opérateurs pseudodifférentiels par de tels fibrés en droites donne lieu à une algébroïde contenant des éléments elliptiques dont l’indice à valeurs dans les réels est donné par une variante de la formule de l’indice d’Atiyah et Singer. En utilisant des idées de Boutet de Monvel et Guillemin, on montre que, sur toute variété symplectique compacte, il est possible d’obtenir le produit étoilé de Lecomte et DeWilde ([3]) (voir aussi la construction de Fedosov ([7]) à partir de l’algébroïde associée au twist des opérateurs de Toeplitz. Cela établit du même coup que la trace définie sur cette algèbre étoilée peut être identifiée avec la trace résiduelle de Wodzicki ([18]) et Guillemin ([10]).

DOI : https://doi.org/10.5802/aif.2060
Classification:  47L80,  53D55
Keywords: deformation quantization, star product, Toeplitz algebra, local line bundle, gerbe, Szeghö projection, contact manifold, index formula, real cohomology.
@article{AIF_2004__54_5_1581_0,
     author = {Melrose, Richard},
     title = {Star products and local line bundles},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1581-1600},
     doi = {10.5802/aif.2060},
     zbl = {1061.47064},
     mrnumber = {2127859},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2004__54_5_1581_0}
}
Melrose, Richard. Star products and local line bundles. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1581-1600. doi : 10.5802/aif.2060. aif.centre-mersenne.org/item/AIF_2004__54_5_1581_0/

[1] R. Beals; P. Greiner Calculus on Heisenberg manifolds, Annals of Mathematics Studies, Tome vol. 119, Princeton University Press, Princeton, NJ, 1988 | MR 953082 | Zbl 0654.58033

[2] L. Boutet de Monvel; V. Guillemin The spectral theory of Toeplitz operators, Ann. of Math. Studies, Tome vol. 99, Princeton University Press, 1981 | MR 620794 | Zbl 0469.47021

[3] M. De Wilde; P.B.A. Lecomte Star-produits et déformations formelles associées aux variétés symplectiques exactes, C.R. Acad. Sci. Paris Sér. I Math, Tome 296 (1983) no. 19, pp. 825-828 | MR 711841 | Zbl 0525.58040

[4] Ch. Epstein; R. Melrose Contact degree and the index of Fourier integral operators, Math. Res. Lett, Tome 5 (1998) no. 3, pp. 363-381 | MR 1637844 | Zbl 0929.58012

[5] C.L. Epstein; R. B. Melrose The Heisenberg algebra, index theory and homology (This became (6) without Mendoza as coauthor)

[6] C.L. Epstein; R. B. Melrose; G. Mendoza The Heisenberg algebra, index theory and homology (in preparation)

[7] B.V. Fedosov Deformation quantization and asymptotic operator representation, Funktsional. Anal. i Prilozhen, Tome 25 (1991) no. 3, pp. 24-36 | MR 1139872 | Zbl 0737.47042

[8] D. Geller Analytic pseudodifferential operators for the Heisenberg group and local solvability, Princeton University Press, Princeton, NJ, 1990 | MR 1030277 | Zbl 0695.47051

[9] V. Guillemin Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys, Tome 35 (1995) no. 1, pp. 85-89 | MR 1346047 | Zbl 0842.58041

[10] V.W. Guillemin A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math, Tome 55 (1985), pp. 131-160 | MR 772612 | Zbl 0559.58025

[11] L. Hörmander The analysis of linear partial differential operators Tome vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985 | MR 404822 | Zbl 0601.35001

[12] V. Mathai; R.B. Melrose; I.M. Singer Fractional analytic index (Submitted) | Zbl 05068853

[13] R.B. Melrose; V. Nistor $K$-theory of $C^*$-algebras of $b$-pseudodifferential operators, Geom. Funct. Anal, Tome 8 (1998), pp. 88-122 | MR 1601850 | Zbl 0898.46060

[14] R.B. Melrose The Atiyah-Patodi-Singer index theorem, A K Peters Ltd., Wellesley, MA, 1993 | MR 1348401 | Zbl 0796.58050

[15] M.K. Murray Bundle gerbes, J. London Math. Soc, Tome 54 (1996), pp. 403-416 | MR 1405064 | Zbl 0867.55019

[16] R.T. Seeley Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math. Chicago, III, 1966) (1967), pp. 288-307 | Zbl 0159.15504

[17] M.E. Taylor Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc, Tome vol. 313, AMS, 1984 | MR 764508 | Zbl 0554.35025

[18] M. Wodzicki Noncommutative residue. I. Fundamentals, $K$-theory, arithmetic and geometry (Moscow, 1984--1986) (Lecture Notes in Math.) (1987), pp. 320-399 | Zbl 0649.58033