A hull that contains no discs

TL;DR

This paper proves that certain Banach algebras' maximal ideal spaces cannot contain high-dimensional, simply coconnected subspaces, leading to the construction of polynomial hulls in complex space without topological discs, strengthening previous results.

Contribution

It demonstrates the nonexistence of certain high-dimensional subspaces in maximal ideal spaces of specific Banach algebras, enabling the construction of polynomial hulls without topological discs.

Findings

Maximal ideal spaces lack compact, locally connected, simply coconnected subspaces of topological dimension ≥ 2.

Existence of a polynomial hull in rac{}^2 containing no topological discs.

Strengthens Stolzenberg's 1963 result on polynomial hulls without analytic 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 $\geq 2$. 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.