A characterization of $C(K)$ among function algebras on a Riemann surface

Abstract

For a compact subset $K$ of a Riemann surface, necessary and sufficient conditions are given for a function algebra containing $A(K)$ to be all of $C(K)$. Using these results, several conditions are given on a complex-valued function $f$ so that the algebra generated by $A(K)$ and $f$ is all of $C(K)$. In particular, the results are applied to a harmonic function $f$ to give sufficient conditions for the algebra generated by $A(K)$ and $f$ to be all of $C(K)$. Also, sufficient conditions are given for the algebra $A(K)$ to be a maximal subalgebra of $C(K)$.