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

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

2011-03-11

arxiv.org/abs/1103.2347v1

Mathematics

Complex Variables

30 pages

Scientific paper

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.

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