Deformation quantization and Borel's theorem in locally convex spaces

Miroslav Engliš¹ ; Jari Taskinen²

¹Mathematics Institute Žitná 25 11567 Praha 1, Czech Republic and Mathematics Institute Na Rybníčku 1 74601 Opava, Czech Republic
²Department of Mathematics and Statistics University of Helsinki P.O. Box 68 FIN-00014 Helsinki, Finland

Studia Mathematica, Tome 180 (2007) no. 1, pp. 77-93

Résumé

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.

Détail

DOI : 10.4064/sm180-1-6

Keywords: nbsp known often construct star product expanding product toeplitz operators asymptotically nbsp series other toeplitz operators multiplied increasing powers planck constant nbsp berezin toeplitz quantization nbsp obtain similar star product which equivalent berezin toeplitz star product nbsp using instead toeplitz operators other suitable mappings compactly supported smooth functions bounded linear operators corresponding hilbert spaces nbsp crucial ingredient proof generalization due colombeau nbsp classical theorem borel existence function prescribed derivatives orders point which reduces proof construction locally convex space enjoying special properties