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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
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##### References:
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