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For a connected real Lie group $G$ we consider the canonical standard-orderedstar product arising from the canonical global symbol calculus based on thehalf-commutator connection of $G$. This star product trivially converges onpolynomial functions on $T^*G$ thanks to its homogeneity. We define a nuclearFr\'echet algebra of certain analytic functions on $T^*G$, for which thestandard-ordered star product is shown to be a well-defined continuousmultiplication, depending holomorphically on the deformation parameter $\hbar$.This nuclear Fr\'echet algebra is realized as the completed (projective) tensorproduct of a nuclear Fr\'echet algebra of entire functions on $G$ with anappropriate nuclear Fr\'echet algebra of functions on $\mathfrak{g}^*$. Thepassage to the Weyl-ordered star product, i.e. the Gutt star product on $T^*G$,is shown to be preserve this function space, yielding the continuity of theGutt star product with holomorphic dependence on $\hbar$.

Convergent Star Products on Cotangent Bundles of Lie Groups