It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions to bounded linear operators on the corresponding Hilbert spaces. A crucial ingredient in the proof is the generalization, due to Colombeau, of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, which reduces the proof to a construction of a locally convex space enjoying some special properties.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-6,
author = {Miroslav Engli\v s and Jari Taskinen},
title = {Deformation quantization and Borel's theorem in locally convex spaces},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {77-93},
zbl = {1125.46004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-6}
}
Miroslav Engliš; Jari Taskinen. Deformation quantization and Borel's theorem in locally convex spaces. Studia Mathematica, Tome 178 (2007) pp. 77-93. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-6/