Projective Freeness of Algebras of Bounded Holomorphic Functions on Infinitely Connected Domains

The algebra $H^{\mathrm{\infty }}(D)$ of bounded holomorphic functions on $D\subset \mathbb{C}$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in $H^{\mathrm{\infty }}(D)$ can be extended in this class of matrices to an invertible square matrix (the generalization of the corona theorem for $H^{\mathrm{\infty }}(D)$). This follows from a new result on the structure of the maximal ideal space of $H^{\mathrm{\infty }}(D)$ asserting that its covering dimension is $2$ and the second \v{C}ech cohomology group is trivial.