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

Abstract

The algebra $H^{\infty }(D)$ 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 $H^{\infty }(D)$ can be extended in this class of matrices to an invertible square matrix (the generalization of the corona theorem for $H^{\infty }(D)$). This follows from a new result on the structure of the maximal ideal space of $H^{\infty }(D)$ asserting that its covering dimension is $2$ and the second Čech cohomology group is trivial.

17 pages

Cited by in corpus (1)

• On Stable Rank of ${\mathbf{H}}^{\infty }$ on Coverings of Finite Bordered Riemann Surfaces