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 realanalytic three-manifolds with boundary. Here we establish a more general 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 functions on a manifold of arbitrary dimension, then the essential set for A has empty interior.

A peak point theorem for uniform algebras on real-analytic varieties

“…We may therefore choose finitely many functions from the ideal so that these functions are real-analytic in a fixed neighborhood N ′ ⊂ N of p and so that the common zero set of these functions is equal to Σ Φ ∩ N ′ . Lemma 2.2 (Lemma 2.2 of [4]). Let M be an m-dimensional differentiable submanifold of R n with boundary of class C 1 .…”

A general method for constructing essential uniform algebras

“…A theorem of John Anderson and the Alexander Izzo [2,Theorem 4.2] classifies all the natural uniform algebras containing the identity function z and generated by C 1 -smooth functions on the closed disc. The example constructed below in the proof of Theorem 1.2 is easily seen to contain the function z and thus shows that this classification does not continue to hold without the smoothness hypothesis.…”

Analytic discs and uniform algebras generated by real-analytic functions

“…An example of Basener [8] on the three-sphere shows that the corresponding statement for three-manifolds is false. However, Anderson, Wermer, and the present author [3], [4], [6], [7] established peak point theorems for uniform algebras generated by real-analytic functions on real-analytic varieties. The latest result along those lines is the following.…”