Convergent star products on cotangent bundles of Lie groups

Abstract

AbstractFor 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 $$T^*G$$ T ∗ G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on $$T^*G$$ T ∗ G , for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $$\hbar $$ ħ . 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 $${\mathfrak {g}}^*$$ g ∗ . The passage to the Weyl-ordered star product, i.e. the Gutt star product on $$T^*G$$ T ∗ G , is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $$\hbar $$ ħ .