Deformation quantization and Borel's theorem in locally convex spaces
Volume 180 / 2007
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 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.
