Topological stable rank of H∞(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∞ (Ω) the Banach algebra of all bounded holomorphic functions on O, 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 Suáreźs theorem that the topological stable rank of H ∞(D) is equal to 2, where 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.