Deformation quantization and Borel's theorem in locally convex spaces

Miroslav Engliš, and Jari Taskinen. "Deformation quantization and Borel's theorem in locally convex spaces." Studia Mathematica 180.1 (2007): 77-93. <http://eudml.org/doc/286603>.

@article{MiroslavEngliš2007,
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.},
author = {Miroslav Engliš, Jari Taskinen},
journal = {Studia Mathematica},
keywords = {Berezin-Toeplitz quantization; Borel theorem; Fréchet space; inductive limit},
language = {eng},
number = {1},
pages = {77-93},
title = {Deformation quantization and Borel's theorem in locally convex spaces},
url = {http://eudml.org/doc/286603},
volume = {180},
year = {2007},
}

TY - JOUR
AU - Miroslav Engliš
AU - Jari Taskinen
TI - Deformation quantization and Borel's theorem in locally convex spaces
JO - Studia Mathematica
PY - 2007
VL - 180
IS - 1
SP - 77
EP - 93
AB - 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.
LA - eng
KW - Berezin-Toeplitz quantization; Borel theorem; Fréchet space; inductive limit
UR - http://eudml.org/doc/286603
ER -