  In this work we consider the Gutt star product viewed as an associative
deformation of the symmetric algebra S^\bullet(g) over a Lie algebra g and
discuss its continuity properties: we establish a locally convex topology on
S^\bullet(g) such that the Gutt star product becomes continuous. Here we have
to assume a mild technical condition on g: it has to be an Asymptotic Estimate
Lie algebra. This condition is e.g. fulfilled automatically for all
finite-dimensional Lie algebras. The resulting completion of the symmetric
algebra can be described explicitly and yields not only a locally convex
algebra but also the Hopf algebra structure maps inherited from the universal
enveloping algebra are continuous. We show that all Hopf algebra structure maps
depend analytically on the deformation parameter. The construction enjoys good
functorial properties.
