×

zbMATH — the first resource for mathematics

Star products and local line bundles. (English) Zbl 1061.47064
Summary: 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 M. DeWilde and P. B. A. Lecomte [C. R. Acad. Sci., Paris, Sér. I 296, 825–828 (1983; Zbl 0525.58040)], see also B. V. Fedosov’s construction in [Funct. Anal. Appl. 25, 184–194 (1991); translation from Funkts. Anal. Prilozh. 25, No. 3, 24–36 (1991; Zbl 0737.47042)]. This also shows that the trace on the star algebra is identified with the residue trace of M. Wodzicki [Lect. Notes Math. 1289, 320–399 (1987; Zbl 0649.58033)] and V. Guillemin [Adv. Math. 55, 131–160 (1985; Zbl 0559.58025)].

MSC:
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
53D55 Deformation quantization, star products
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] R. Beals & P. Greiner, Calculus on Heisenberg manifolds, Annals of Mathematics Studies vol. 119, Princeton University Press, 1988 · Zbl 0654.58033
[2] L. Boutet de Monvel & V. Guillemin, The spectral theory of Toeplitz operators, Ann. of Math. Studies vol. 99, Princeton University Press, 1981 · 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 Math296 (1983) no. p. 825-828 · Zbl 0525.58040
[4] Ch. Epstein & R. Melrose, Contact degree and the index of Fourier integral operators, Math. Res. Lett5 (1998) no. p. 363-381 · 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 Prilozhen25 (1991) no. p. 24-36 · Zbl 0737.47042
[8] D. Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, Princeton University Press, 1990 · Zbl 0695.47051
[9] V. Guillemin, Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys35 (1995) no. p. 85-89 · Zbl 0842.58041
[10] V.W. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. Math55 (1985) p. 131-160 · Zbl 0559.58025
[11] L. Hörmander, The analysis of linear partial differential operators, vol. 3, Springer-Verlag, 1985 · Zbl 0601.35001
[12] V. Mathai, R.B. Melrose & I.M. Singer, Fractional analytic index, Submitted · Zbl 1115.58021
[13] R.B. Melrose & V. Nistor \(, K\)-theory of \(C^*\)-algebras of \(b\)-pseudodifferential operators, Geom. Funct. Anal8 (1998) p. 88-122 · Zbl 0898.46060
[14] R.B. Melrose, The Atiyah-patodi-Singer index theorem, A K Peters Ltd., 1993 · Zbl 0796.58050
[15] M.K. Murray, Bundle gerbes, J. London Math. Soc54 (1996) p. 403-416 · Zbl 0867.55019
[16] R.T. Seeley, Complex powers of an elliptic operator, Amer. Math. Soc., 1967, p. 288-307 · Zbl 0159.15504
[17] M.E. Taylor, Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc vol. 313, AMS, 1984 · Zbl 0554.35025
[18] M. Wodzicki, Noncommutative residue. I. fundamentals, Lecture Notes in Math., Springer, 1987, p. 320-399 · Zbl 0649.58033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.