  Starting from formal deformation quantization we use an explicit formula for
a star product on the Poincar\'e disk D_n to introduce a Fr\'echet topology
making the star product continuous. To this end a general construction of
locally convex topologies on algebras with countable vector space basis is
introduced and applied. Several examples of independent interest are
investigated as e.g. group algebras over finitely generated groups and infinite
matrices. In the case of the star product on D_n the resulting Fr\'echet
algebra is shown to have many nice features: it is a strongly nuclear K\"othe
space, the symmetry group SU(1, n) acts smoothly by continuous automorphisms
with an inner infinitesimal action, and evaluation functionals at all points of
D_n are continuous positive functionals.
