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

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.