Abstract
It is shown that if $A$ is a uniform algebra generated by a family $\Phi$ of complex-valued $C^1$ functions on a compact $C^1$ manifold-with-boundary $M$, the maximal ideal space of $A$ is $M$, and $E$ is the set of points where the differentials of the functions in $\Phi$ fail to span the complexified cotangent space to $M$, then $A$ contains every continuous function on $M$ that vanishes on $E$. This answers a 45-year-old question of Michael Freeman who proved the special case in which the manifold $M$ is two-dimensional. More general forms of the theorem are also established. The results presented strengthen results due to several mathematicians.
Full Article