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On the Banach-Stone theorem for algebras of holomorphic germs

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Abstract

Given two balanced compact subsets K and L of two Banach spaces X and Y respectively such that every continuous m-homogeneous polynomial on \(X^{**}\) and on \(Y^{**}\) is approximable, for all \(m\in \mathbb {N}\), we characterize when the algebras of holomorphic germs \(\mathcal {H}(K)\) and \(\mathcal {H}(L)\) are topologically algebra isomorphic. This happens if and only if the polynomial hulls of K and L on their respective biduals are biholomorphically equivalent.

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Acknowledgments

Part of this paper was written during the visit of D. M. Vieira to the Universidad de Valencia, and she wants to thank the Departamento de Análisis Matemático of the Universidad de Valencia and its members for their kind hospitality.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100, Burjasot, Valencia, Spain

    Domingo García & Manuel Maestre

  2. Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, CEP 05508-090, Brazil

    Daniela M. Vieira

Authors
  1. Domingo García
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  2. Manuel Maestre
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  3. Daniela M. Vieira
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Correspondence to Domingo García.

Additional information

Dedicated to Professor José Bonet on his 60th birthday.

The D. García and M. Maestre authors were supported by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013. The D. M. Vieira was supported by FAPESP Proc. 2014/07373-0 (Brazil).

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García, D., Maestre, M. & Vieira, D.M. On the Banach-Stone theorem for algebras of holomorphic germs. RACSAM 111, 223–230 (2017). https://doi.org/10.1007/s13398-016-0289-z

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  • DOI: https://doi.org/10.1007/s13398-016-0289-z

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