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Revista Matemática Complutense

, Volume 31, Issue 3, pp 651–672 | Cite as

Algebras of symmetric holomorphic functions of several complex variables

  • Richard M. Aron
  • Javier FalcóEmail author
  • Domingo García
  • Manuel Maestre
Article

Abstract

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open and bounded set with \(\overline{U}\subset \varOmega \)), we study the subalgebra \({\mathcal {B}}_{g}\) of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk.

Keywords

Symmetric holomorphic functions Fibers Algebras of holomorphic functions Several complex variables Symmetrized polydisk 

Mathematics Subject Classification

Primary 32A38 Secondary 30H05 05E05 

Notes

Acknowledgements

The authors would like to thank the referee for remarks that lead to a clarification of Theorem 2 and Corollary 4.

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA
  2. 2.Departamento de Análisis MatemáticoUniversidad de ValenciaBurjasotSpain

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