A hull that contains no discs

Abstract

It is shown that if $A$ is a unital commutative Banach algebra with a dense set of invertible elements, then the maximal ideal space of $A$ contains no compact, locally connected, simply coconnected subspace of topological dimension $\ge 2$. As a consequence, the existence of a compact set in ${\mathbb{C}}^2$ with a nontrivial polynomial hull that contains no topological discs is obtained. This strengthens the celebrated result of Stolzenberg from 1963 that there exists a nontrivial polynomial hull that contains no analytic discs, and it answers a question stated in the literature 10 years ago by Dales and Feinstein but considered much earlier.