Peak Point Theorems for Uniform Algebras on Smooth Manifolds

Abstract: It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if each point of X is a peak point for A, then A = C(X). This peak point conjecture was disproved by Brian Cole in 1968. However, Anderson and Izzo showed that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth two-manifolds with boundary. The corresponding assertion for smooth three-manifolds is false, but Anderson, Izzo, and Wermer established a peak point theorem for polynomial approximation on real-analytic three-manifolds with boundary. Here we establish a more gen-eral peak point theorem for real-analytic three-manifolds with boundary analogous to the two-dimensional result. We also show that if A is a counterexample to the peak point conjecture generated by smooth func-tions on a manifold of arbitrary dimension, then the essential set for A has empty interior. 1