Topological Stable Rank of $H^\infty(\Omega)$ for Circular Domains $\Omega$

TL;DR

This paper determines that the topological stable rank of the algebra of bounded holomorphic functions on circular domains is 2, extending known results from the unit disk to these more complex domains.

Contribution

It establishes the topological stable rank of $H^ abla(\Omega)$ for circular domains and symmetric domains, generalizing Suarez's theorem from the unit disk to these cases.

Findings

Topological stable rank of $H^ abla(\Omega)$ is 2 for circular domains.

For symmetric domains, the Bass and topological stable ranks of $H^ abla_ ext{R}(\Omega)$ are 2.

Extension of Suarez's theorem to more general domains.

Abstract

Let $\Omega$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebra of all bounded holomorphic functions on $\Omega$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(\Omega)$ is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the unit disk. We also show that for domains symmetric to the real axis, the Bass and topological stable ranks of the real symmetric algebra $H^\infty_\R(\Omega)$ are 2.