  Looking for the universal covering of the smooth non-commutative torus leads
to a curve of associative multiplications on the space $\Cal O_M'(\Bbb
R^{2n})\cong \Cal O_C(\Bbb R^{2n})$ of Laurent Schwartz which is smooth in the
deformation parameter $\hbar$. The Taylor expansion in $\hbar$ leads to the
formal Moyal star product. The non-commutative torus and this version of the
Heisenberg plane are examples of smooth *-algebras: smooth in the sense of
having many derivations. A tentative definition of this concept is given.
