A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two‐Manifolds

We establish the peak point conjecture for uniform algebras generated by smooth functions on two‐manifolds: if A is a uniform algebra generated by smooth functions on a compact smooth two‐manifold M , such that the maximal ideal space of A is M , and every point of M is a peak point for A , then A = C ( M ). We also give an alternative proof in the case when the algebra A is the uniform closure P ( M ) of the polynomials on a polynomially convex smooth two‐manifold M lying in a strictly pseudoconvex hypersurface in C n .