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

Let $Ω$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(Ω)$ the Banach algebra of all bounded holomorphic functions on $Ω$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(Ω)$ 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 $H^\infty_\R(Ω)$ are 2.

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