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

Abstract

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, 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.

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.…”

“…It has, however, been strengthened by Swarup Ghosh [16] by replacing the peak point hypothesis by the the hypothesis that each point of the maximal ideal space is isolated in the topology inherited from the normed dual A * of A. The real-analytic peak point theorems in dimension three given in [3] and [6] have also been similarly strengthened by Ghosh [16], [17].…”

A general method for constructing essential uniform algebras

“…A counterexample to this peak-point conjecture was produced by Brian Cole in 1968 [9] (or see [8,Appendix], or [29, Chapter 3, Section 19]), and other counterexamples have been given since then with a variety of additional properties, and including examples which are generated by smooth functions on manifolds (see, for example, [6,12,13,14,22,23,30]). Nevertheless, Anderson and Izzo [1,2,3] and Anderson, Izzo, and Wermer [4,5] have established peak point theorems under certain smoothness hypotheses. One of those results, which we state here, asserts that certain uniform algebras can never be essential.…”

Localization for uniform algebras generated by real-analytic functions

“…Since A satisfies the countable approximation property if each function in C(K) is locally uniformly approximable by functions in A, the above theorem answers the question from [7] mentioned above in the special case of uniform algebras generated by real-analytic functions. The choice of this setting was suggested by work of the authors and Wermer ([2], [3], [4], [5], [6]) on the peak-point conjecture. This conjecture concerned a possible characterization of C(X) which was shown to be false in general by Cole [8] but is true in a number of special cases of interest.…”