Equality of seven fundamental sets connected with A(K): analytic capacity-free proofs

TL;DR

This paper proves the equivalence of seven fundamental sets related to a compact set in the complex plane using only classical complex analysis, avoiding advanced theories like analytic capacity.

Contribution

It provides a new proof of these equivalences relying solely on classical complex analytic tools, without using the theory of analytic capacity or related advanced concepts.

Findings

Equivalence of boundary and algebraic properties for A(K)

Bounded approximate identity characterization of boundary points

No use of advanced analytic capacity theories in proofs

Abstract

Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let z be in K, and let M be the maximal ideal of functions in A(K) that are 0 at z. Using only facts from classical complex analytic function theory, and without using any results from the theory of analytic capacity, we prove the following: z is an element of bK iff z is a peak point for A(K) iff z belongs to the Shilov boundary for A(K) iff z belongs to the Bishop minimal boundary for A(K) iff M has a bounded approximate identity iff the Bishop one-quarter - three-quarter property holds at z iff z is a strong boundary point for A(K). More specifically, the only results used in all proofs are from classical complex analytic function theory, properties of open connected sets in the complex plane, the Caratheodory extension theorem, the Riemann mapping theorem, Euler's formula, Rudin's estimates for finite complex products, properties of linear fractional transformations, the alpha root function in the complex plane, some new and striking Jordan curve constructions (Jordan kissing paths), and the Cohen factorization theorem. No results are used from the theory of analytic capacity, the theory of representing or annihilating measures, Dirichlet algebra theory, Choquet boundary theory, or the Walsh-Lebesgue theorem.