Polynomial approximation on three-dimensional real-analytic submanifolds of C-n

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, it was recently shown by Anderson and Izzo that the peak point conjecture does hol d for uniform algebras generated by smooth functions on smooth two-manifold s with boundary. Although the corresponding assertion for smooth three-mani folds is false, we establish a peak point theorem for real-analytic three-m anifolds with boundary.