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Peak Point Theorems for Uniform Algebras on Smooth Manifolds John T. Anderson and Alexander J. Izzo
 

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 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. 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 complex-valued continuous functions on X) that is closed in the
supremum norm, contains the constants, and separates the points of X. A central problem

  

Source: Anderson, John T. - Department of Mathematics and Computer Science, College of the Holy Cross

 

Collections: Mathematics