Uniform approximation on Riemann surfaces by holomorphic and harmonic functions

Abstract

Let $K$ be a compact subset of an open Riemann surface. We prove that if $L$ is a peak set for $A(K)$, then $A(K)|L=A(L).$ We also prove that if $E$ is a compact subset of $K$ with no interior such that each component of $E^c$ intersects $K^c$, then $A(K)|E$ is dense in $C(E)$. One consequence of the latter result is a characterization of the real-valued continuous functions that when adjoined to $A(K)$ generate $C(K)$.