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

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Abstract

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 \(\mathbb{D}\subset \mathbb{C}\) 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 \(H_{\mathrm{comp}}^{\infty}(A)\) of holomorphic functions on \(\mathbb{D}\) 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada

    Alexander Brudnyi

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  1. Alexander Brudnyi
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Correspondence to Alexander Brudnyi.

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Research supported in part by NSERC.

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Brudnyi, A. Banach-valued holomorphic functions on the maximal ideal space of H ∞ . Invent. math. 193, 187–227 (2013). https://doi.org/10.1007/s00222-012-0426-z

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