Topological stable rank of H ∞(Ω) for circular domains Ω

Abstract

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ∞(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 ∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ∞(\( \mathbb{D} \)) is equal to 2, where \( \mathbb{D} \) is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H ∞ℝ (Ω) are 2.