The Choquet boundary for uniform algebras

Abstract

By a uniform algebra (or function algebra) in C _c(Y) (Y compact Hausdorff) we mean any uniformly closed subalgebra of C _c(Y) which contains the constant functions and separates points of Y. For metrizable Y, the Choquet boundary of a uniform algebra A has a particularly simple description (Bishop [8]): It consists of the peak points for A, i.e., of those y in Y for which there exists a function f in A with the property that |f(x)| < |f(y)| if x ≠ y. This result is a special case of a characterization (for arbitrary Y) due to Bishop and de Leeuw [9], which is the main theorem of this section.