Estimates for the Corona Theorem on $H^{\infty}_{\mathbb{I}}(\D)$

Abstract: Let $\mathbb{I}$ be a proper ideal of $H^{\infty}(\D)$. We prove the corona theorem for infinitely many generators on the algebra $H^{\infty}_{\mathbb{I}}$ in which the corona theorem for finitely many functions is known to hold. This settles the conjecture of Ryle \cite{ryle1}. We also provide the estimates for corona solutions. Moreover, we prove a generalized Wolff's Ideal Theorem for this sub-algebra.