  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 $T^*G$ thanks to its homogeneity. We define a nuclear
Fr\'echet algebra of certain analytic functions on $T^*G$, for which the
standard-ordered star product is shown to be a well-defined continuous
multiplication, depending holomorphically on the deformation parameter $\hbar$.
This nuclear Fr\'echet algebra is realized as the completed (projective) tensor
product of a nuclear Fr\'echet algebra of entire functions on $G$ with an
appropriate nuclear Fr\'echet algebra of functions on $\mathfrak{g}^*$. The
passage 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 the
Gutt star product with holomorphic dependence on $\hbar$.
