The Bass and topological stable ranks of (H_ℝ ^∞(D) and A_ℝ(D)

In this note we prove that the Bass stable rank of H_ℝ ^∞(D) is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of H^∞_ℝ(D) is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of A_ℝ(D) is two and the topological stable rank of A _ℝ(D) is two, settling a conjecture by the second author. We also present a ∂̄-free proof of the second author's characterization of the reducible pairs in A_ℝ(D).