  $AdS_3$ space-time admits a foliation by two-dimensional twisted conjugacy
classes, stable under the identification subgroup yielding the non-rotating
massive BTZ black hole. Each leaf constitutes a classical solution of the
space-time Dirac-Born-Infeld action, describing an open D-string in $AdS_3$ or
a D-string winding around the black hole. We first describe two nonequivalent
maximal extensions of the non-rotating massive BTZ space-time and observe that
in one of them, each D-string worldsheet admits an action of a two-parameter
subgroup ($\ca \cn$) of $\SL$. We then construct non-formal, $\ca
\cn$-invariant, star products that deform the classical algebra of functions on
the D-string worldsheets and on their embedding space-times. We end by giving
the first elements towards the definition of a Connes spectral triple on
non-commutative $AdS$ space-times.
