  We define a universal deformation formula (UDF) for the actions of the affine
group on Frechet algebras. More precisely, starting with any associative
Frechet algebra which the affine group acts on in a strongly continuous and
isometrical manner, the UDF produces a family of topological associative
algebra structures on the space of smooth vectors of the action deforming the
initial product. The deformation field obtained is based over an infinite
dimensional parameter space naturally associated with the space of
pseudo-differential operators on the real line. This note also presents some
geometrical aspects of the UDF and in particular its relation with hyperbolic
geometry.
