Uniform approximation on manifolds


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.


Alexander J. Izzo

Department of Mathematics, Bowling Green State University, Bowling Green, OH 43403