A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

Abstract

We establish the peak point conjecture for uniform algebras generated by smooth functions on twomanifolds: if A is a uniform algebra generated by smooth functions on a compact smooth two-manifold M, such that the maximal ideal space of A is M, and every point of M is a peak point for A, then A = C(M). We also give an alternative proof in the case when the algebra A is the uniform closure P (M) of the polynomials on a polynomially convex smooth two-manifold M lying in a strictly pseudoconvex hypersurface in C n .

Peak point theorems for uniform algebras on smooth manifolds

“…In 2001 Anderson and Izzo [1] established a peak point theorem for uniform algebras on two-dimensional manifolds: if A is a uniform algebra generated by C 1 functions on a compact C 1 two-dimensional manifold with boundary M such that the maximal ideal space of A is M and each point of M is a peak point for A, then A = C(M ). The corresponding assertion for three-dimensional manifolds is false: there is a counterexample to the peak point conjecture due to Izzo [8] generated by C ∞ functions on a smooth solid torus, and a counterexample generated by C ∞ functions on the 3-sphere can easily be obtained from Basener's example (see [1]).…”

“…The corresponding assertion for three-dimensional manifolds is false: there is a counterexample to the peak point conjecture due to Izzo [8] generated by C ∞ functions on a smooth solid torus, and a counterexample generated by C ∞ functions on the 3-sphere can easily be obtained from Basener's example (see [1]). Nevertheless Anderson, Izzo, and Wermer [2] proved a peak point theorem for polynomial approximation on real-analytic three-dimensional submanifolds of C n .…”

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

“…Note that since X satisfies (i) and (ii), Lemma 2.1 shows that ∂X also satisfies (i) and (ii). Since ∂X is a two-manifold, the peak point theorem for two-manifolds [2] shows that P (∂X) = C(∂X). Moreover, Lemma 2.1 implies that Z is polynomially convex.…”

“…Their work relies on methods of Michael Freeman [6], who studied algebras generated by smooth functions on a two-manifold, generalizing the work of Wermer on polynomially convex disks mentioned above. As a special case of the results of [2], one obtains the fact that if K = M is a two-manifold in C n satisfying (i) and (ii), then P (M ) = C(M ). The purpose of this paper is to prove the following analogous result on approximation on three-manifolds:…”

“…Izzo's paper also includes an example of this type where X is a smooth solid torus in ∂B 5 . In a positive direction, Anderson and Izzo [2] have recently established the peak point conjecture for uniform algebras generated by smooth functions on a twomanifold (with boundary). Their work relies on methods of Michael Freeman [6], who studied algebras generated by smooth functions on a two-manifold, generalizing the work of Wermer on polynomially convex disks mentioned above.…”