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 \(C(K)\).