Topology of the Maximal Ideal Space of $H^{\mathrm{\infty }}$

We study the structure of the maximal ideal space $M(H^{\mathrm{\infty }})$ of the algebra $H^{\mathrm{\infty }}=H^{\mathrm{\infty }}(\text{Di})$ of bounded analytic functions defined on the open unit disk $\text{Di}\subset \text{Co}$. Based on the fact that $dim
M(H^{\infty})=2$weprovefor$H^{\infty}$ the matrix-valued corona theorem. Our results heavily rely on the topological construction describing maximal ideal spaces of certain algebras of continuous functions defined on the covering spaces of compact manifolds.