# Star products and local line bundles

Richard Melrose^{[1]}

- [1] Massachusetts Institute of Technology, Department of Mathematics (USA)

Annales de l’institut Fourier (2004)

- Volume: 54, Issue: 5, page 1581-1600
- ISSN: 0373-0956

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topMelrose, Richard. "Star products and local line bundles." Annales de l’institut Fourier 54.5 (2004): 1581-1600. <http://eudml.org/doc/116153>.

@article{Melrose2004,

abstract = {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]).},

affiliation = {Massachusetts Institute of Technology, Department of Mathematics (USA)},

author = {Melrose, Richard},

journal = {Annales de l’institut Fourier},

keywords = {deformation quantization; star product; Toeplitz algebra; local line bundle; gerbe; Szeghö projection; contact manifold; index formula; real cohomology; Szegö projection},

language = {eng},

number = {5},

pages = {1581-1600},

publisher = {Association des Annales de l'Institut Fourier},

title = {Star products and local line bundles},

url = {http://eudml.org/doc/116153},

volume = {54},

year = {2004},

}

TY - JOUR

AU - Melrose, Richard

TI - Star products and local line bundles

JO - Annales de l’institut Fourier

PY - 2004

PB - Association des Annales de l'Institut Fourier

VL - 54

IS - 5

SP - 1581

EP - 1600

AB - 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]).

LA - eng

KW - deformation quantization; star product; Toeplitz algebra; local line bundle; gerbe; Szeghö projection; contact manifold; index formula; real cohomology; Szegö projection

UR - http://eudml.org/doc/116153

ER -

## References

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