Convergent Star Products on Cotangent Bundles of Lie Groups

Abstract

For a connected real Lie group $G$ we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of $G$. This star product trivially converges on polynomial functions on $T^{\ast }G$ thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on $T^{\ast }G$, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $\hslash$. This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on $G$ with an appropriate nuclear Fréchet algebra of functions on ${\mathfrak{g}}^{\ast }$. The passage to the Weyl-ordered star product, i.e. the Gutt star product on $T^{\ast }G$, is shown to be preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $\hslash$.

44 pages