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

TL;DR

This paper characterizes when functions on planar compacta generate specific algebras and explores conditions for their holomorphic injective extensions, revealing surprising examples and boundary behavior insights.

Contribution

It provides necessary and sufficient conditions for algebra generation and injective holomorphic extension of functions on planar compacta, including new examples and boundary criteria.

Findings

Examples of non-polynomial convex compacta with homeomorphic functions lacking inverse in P(K)

Conditions for functions to admit holomorphic injective extensions

Results on continuous logarithms on punctured compacta

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.