Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^\infty

TL;DR

This paper investigates Banach-valued holomorphic functions on the maximal ideal space of H^^\u2206, establishing vanishing cohomology, solving a corona problem, and characterizing the maximal ideal space of related function algebras.

Contribution

It introduces new results on the structure of Banach-valued holomorphic functions, including vanishing cohomology and a corona theorem, on the maximal ideal space of H^^.

Findings

Vanishing cohomology for sheaves of Banach-valued holomorphic functions.

Solution to a Banach-valued corona problem for H^.

Homeomorphism between maximal ideal space of H_{ m comp}^^ (A) and the product of maximal ideal spaces.

Abstract

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H^\infty of bounded holomorphic functions on the unit disk D\subset 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^\infty, prove that the maximal ideal space of the algebra H_{\rm comp}^\infty (A) of holomorphic functions on $\Di$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H^\infty and A.