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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 hold for uniform algebras generated by smooth functions on smooth twomanifolds with boundary. Although the corresponding assertion for smooth three-manifolds is false, we establish a peak point theorem for real-analytic three-manifolds with boundary.

Polynomial approximation on three-dimensional real-analytic submanifolds of 𝐂ⁿ