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

Abstract

We study the structure of the maximal ideal space $M(H^{\infty })$ of the algebra $H^{\infty}=H^{\infty}(\Di)$ of bounded analytic functions defined on the open unit disk $\Di\subset\Co$. Based on the fact that $dimM(H^{\infty })=2$ we prove for $H^{\infty }$ 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.