Non-formal deformation quantization of Frechet-Poisson algebras: The Heisenberg Lie algebra case

摘要

The purpose of this paper is to give examples of deformation quantization of Poisson algebras with convergent star product in the Frechet category. The problem of deformation quantizations initiated by [BFFLS] has been developed extensively in recent years, and there are many interesting papers on formal deformation quantization: i.e.. deformation quantizations of Poisson algebras with a formal power series in deformation parameter. One can ask whether the formal parameter expansion converges. In this spirit, Rieffel [R] introduced a notion of strict deformation quantization: deformation quantization with a convergent product for C*-algebras, This suggested to us the possiblity of denning deformation quantization with convergent product for suitable categories of algebras. In particular, we have denned a notion of deformation quantization corresponding to Rieffel's work [R] in the Frechet category and have showed that a class of entire functions with growth conditions proposed by Gel'fand and Shilov [GS] behave well for deformation quantization, As shown in [0MMY1], it seems necessary to define the notion of deformation quantization in the Frechet category rather than in the C*-algebra framework if we want to exponentiate quadratic forms. The results in this paper have been announced in [0MMY3] without proof, and we supply complete proofs in this paper.