Deformation quantization and Borel's theorem in locally convex spaces

Tom 180 / 2007

Miroslav Engliš, Jari Taskinen Studia Mathematica 180 (2007), 77-93 MSC: Primary 46A13; Secondary 53D55, 26B05, 47B35. DOI: 10.4064/sm180-1-6


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.


  • Miroslav EnglišMathematics Institute
    Žitná 25
    11567 Praha 1, Czech Republic
    Mathematics Institute
    Na Rybníčku 1
    74601 Opava, Czech Republic
  • Jari TaskinenDepartment of Mathematics and Statistics
    University of Helsinki
    P.O. Box 68
    FIN-00014 Helsinki, Finland

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