Representing Measures for Points in a Uniform Algebra

Recent years have seen much effort put into attempts to develop an "abstract" function theory. In the complex domain, this has led to the study of the structure of uniform algebras. A uniform algebra 31 is a family of continuous complex-valued functions on a compact Hausdorff space X, which contains the function 1, which is closed with respect to the algebraic operations of addition and multiplication by complex scalars, which is topologically closed in the uniform norm, and which distinguishes points of X. Succinctly, 91 is a closed separating unitary subalgebra of the Banach algebra C(X) of all continuous complex-valued functions on X. Standard examples are obtained by taking X to be a compact subset of complex Euclidean space C and SI to be the closed subalgebra of C(X) generated by the constants and the coordinate functions on C. Some of the most suggestive ideas in the theory of uniform algebras come from a paper of Gleason [l ]. He calls elements x and y in X equivalent if