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2007 | 180 | 1 | 77-93
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Deformation quantization and Borel's theorem in locally convex spaces

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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.
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  • Mathematics Institute, Žitná 25, 11567 Praha 1, Czech Republic
  • Mathematics Institute, Na Rybníčku 1, 74601 Opava, Czech Republic
  • Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 Helsinki, Finland
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm180-1-6
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