  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.
