Stein-like theory for Banach-valued holomorphic functions on the maximal ideal space of algebra of bounded holomorphic functions

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 D ⊂ C 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 of holomorphic functions on D
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.