$C^\infty$ Functions on the Stone-\v{C}ech Compactification of the Integers

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 $\Bbb Z$.