On the closure of the sum of two uniform algebras on compact sets in \(\mathbb C\)
Let \(X\subseteq \mathbb C\) be a compact set without interior points and whose complement is disconnected. The polynomial convex hull of \(X\) is denoted by \(\widehat X\). For a homeomorphism \(\psi\) of \(\mathbb C\) let \(Z(X,\psi)\) be the closure in \(C(X)\) of the set of functions of the form \(p+(h\circ \psi)\) where \(p\) is a polynomial and \(h\) a rational function whose poles belong to the bounded components of \(\mathbb C\setminus\psi(X)\). The author shows the following result. Suppose that \(\psi\) is an orientation-preserving diffeomorphism of \(\mathbb C\) and that each point of the set \[ X\cap \overline{\left(\widehat X\right)^\circ} \] is a boundary point for some bounded component of \(\mathbb C\setminus X\). Then \(Z(X,\psi)=C(X)\). Several nice examples round up the paper.
scientific article; zbMATH DE number 5980744