 
Summary: Peak Point Theorems for Uniform Algebras on Smooth Manifolds
John T. Anderson and Alexander J. Izzo
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 twomanifolds with boundary. The corresponding assertion
for smooth threemanifolds is false, but Anderson, Izzo, and Wermer
established a peak point theorem for polynomial approximation on real
analytic threemanifolds with boundary. Here we establish a more gen
eral peak point theorem for realanalytic threemanifolds with boundary
analogous to the twodimensional 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. Introduction
Let A be a uniform algebra on a compact metric space X. That is, A is a subalgebra
of C(X) (the algebra of all complexvalued continuous functions on X) that is closed in the
supremum norm, contains the constants, and separates the points of X. A central problem
