Banach-valued holomorphic functions on the maximal ideal space of H∞

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H∞ of bounded holomorphic functions on the unit disk \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}\subset \mathbb{C}$\end{document} with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H∞, prove that the maximal ideal space of the algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\mathrm{comp}}^{\infty}(A)$\end{document} of holomorphic functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{D}$\end{document} with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H∞ and A.