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March 25, 2015 By

Banach-valued holomorphic functions on the maximal ideal space of $H^\infty$

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (4) 2011, pp. 97–106

December 30, 2011

Alexander Brudnyi, Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4; e-mail: albru@math.ucalgary.ca

Abstract/Résumé:

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 \(\mathbf{D}\subset\mathbf{C}\) with pointwise multiplication and supremum norm. In particular, we establish the vanishing of the cohomology of 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 \(H_{\operatorname{comp}}^\infty (A)\) of holomorphic functions on \(\mathbf{D}\) with relatively compact images in a commutative unital complex Banach algebra \(A\) is homeomorphic to the direct product of maximal ideal spaces of \(H^\infty\) and \(A\). All proofs are presented in arXiv:1103.2347v1.

Nous étudions des fonctions holomorphes à valeurs dans un espace de Banach et définies sur des sous ensembles ouverts de l’espace idéal maximal de l’algèbre de Banach \(H^\infty\) des fonctions holomorphes bornées sur le disque unité \(\mathbf{D}\subset\mathbf{C}\) munies de la multiplication ponctuelle et de la norme du supremum. En particulier, nous établissons que la cohomologie des faisceaux des germes de ces fonctions est nulle et, par le biais de la résolution d’un problème de type corona pour \(H^\infty\) à valeurs dans un espace de Banach, montrons que l’espace idéal maximal de l’algèbre \(H_{\operatorname{comp}}^\infty (A)\) des fonctions holomorphes sur \(\mathbf{D}\) et à image relativement compacte dans une algèbre complexe commutative unitale de Banach \(A\) est homéomorphe au produit direct des espaces idéaux maximaux de \(H^\infty\) et \(A\). Toutes les preuves sont présentées dans arXiv:1103.2347v1.

Keywords: $bar ∂$-equation, bounded holomorphic function, maximal ideal space, slice algebra
AMS Subject Classification: ${H]^p$-classes 30D55

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