DEFORMATION QUANTIZATION AND BOREL'S THEOREM IN LOCALLY CONVEX SPACES

Abstract. 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 another 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 suitable other assignments from compactly sup-ported 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. 1.Introduction and background Let be a domain in Cn = R2n equipped with a Poisson bracket f; g, i.e. a rst order bidierential operato