Holomorphic injective extensions of functions in P(K) and algebra generators

Abstract

We present necessary and sufficient conditions on planar compacta $K$ and continuous functions $f$ on $K$ in order that $f$ generates the algebras $P(K), R(K), A(K)$ or $C(K)$. We also unveil quite surprisingly simple examples of non-polynomial convex compacta $K\subseteq\mathbb C$ and $f\in P(K)$ with the property that $f \in P(K)$ is a homeomorphism, but for which $f^{-1}\notin P(f(K))$. As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull $\widehat K$. On the other hand, it will be shown that the restriction $f^*|_G$ of the Gelfand-transform $f^*$ of an injective function $f\in P(K)$ is injective on every regular, bounded complementary component $G$ of $K$. A necessary and sufficient condition in terms of the behaviour of $f$ on the outer boundary of $K$ is given in order $f$ admits a holomorphic injective extension to $\widehat K$. We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.