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Let K be any compact set in the complex plane that has a connectedcomplement, let A(K) be the uniforn algebra of all continuous complex functionson K that are holomorphic on the interior of K, let bK be the topologicalboundary of K, let z be in K, and let M be the maximal ideal of functions inA(K) that are 0 at z. Using only facts from classical complex analytic functiontheory, and without using any results from the theory of analytic capacity, weprove the following: z is an element of bK iff z is a peak point for A(K) iff zbelongs to the Shilov boundary for A(K) iff z belongs to the Bishop minimalboundary for A(K) iff M has a bounded approximate identity iff the Bishopone-quarter - three-quarter property holds at z iff z is a strong boundarypoint for A(K). More specifically, the only results used in all proofs are fromclassical complex analytic function theory, properties of open connected setsin the complex plane, the Caratheodory extension theorem, the Riemann mappingtheorem, Euler's formula, Rudin's estimates for finite complex products,properties of linear fractional transformations, the alpha root function in thecomplex plane, some new and striking Jordan curve constructions (Jordan kissingpaths), and the Cohen factorization theorem. No results are used from thetheory of analytic capacity, the theory of representing or annihilatingmeasures, Dirichlet algebra theory, Choquet boundary theory, or theWalsh-Lebesgue theorem.

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