  We construct an algebra $A=\ell^{\infty \infty}({\Bbb Z})$ of smooth
functions which is dense in the pointwise multiplication algebra
$\ell^\infty({\Bbb Z})$ of sup-norm bounded functions on the integers $\Bbb Z$.
The algebra $A$ properly contains the sum of the algebra
$A_c=\ell_c^\infty({\Bbb Z})$ and the ideal ${\cal S}({\Bbb Z})$, where $A_c$
is the algebra of finite linear combinations of projections in
$\ell^\infty({\Bbb Z})$ and ${\cal S}({\Bbb Z})$ is the pointwise
multiplication algebra of Schwartz functions. The algebra $A$ is characterized
as the set of functions whose "first derivatives" vanish rapidly at each point
in the Stone-${\check {\rm C}}$ech compactification of $\Bbb Z$.
