The Bass and Topological Stable Ranks of $H^\infty_\R(\D)$ and $A_\R(\D)$

Abstract

In this note we prove that the Bass stable rank of $H^\infty_\R(\D)$ 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 $H^\infty_\R(\D)$ 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 $\bar{\partial }$-free proof of the second author's characterization of the reducible pairs in $A_\R(\D)$.

19 pages, to appear in J. Reine Angew. Math

References in corpus (1)

• A Note about Stabilization in $A_\R(\D)$

Cited by in corpus (3)

• Spectral Characteristics and Stable Ranks for the Sarason Algebra $H^{\infty }+C$
• Topological Stable Rank of $H^{\infty }(\Omega )$ for Circular Domains $\Omega$
• Reducibility of invertible tuples to the principal component in commutative Banach algebras