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The Spectra of Algebras of Group-Symmetric Functions

Part of: Linear function spaces and their duals Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  29 November 2018

Domingo García ,
Manuel Maestre  and
Ignacio Zalduendo
Domingo García
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Valencia, Spain (domingo.garcia@uv.es; manuel.maestre@uv.es)
Manuel Maestre
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Valencia, Spain (domingo.garcia@uv.es; manuel.maestre@uv.es)
Ignacio Zalduendo
Affiliation:
Universidad Torcuato Di Tella. Av. Figueroa Alcorta 7350 (C1428BCW), Buenos Aires, Argentina (izalduendo@utdt.edu)
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Abstract

In the study of the spectra of algebras of holomorphic functions on a Banach space E, the bidual E″ has a central role, and the spectrum is often shown to be locally homeomorphic to E″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f). It is natural to attempt a characterization in terms of the space of orbits E″/~ obtained from E″ through the action of the group, so we pursue this approach here and introduce an analytic structure on the spectrum in some situations. In other situations we encounter some obstacles: in some cases, the lack of structure of E″/~ itself; in others, problems of weak continuity and non-approximability of functions in the algebra. We also define a convolution operation related to the spectrum.

Keywords

symmetric polynomialgroup of operatorsholomorphic function

MSC classification

Primary: 46G20: Infinite-dimensional holomorphy
Secondary: 46E50: Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 609 - 623
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Alencar, R., Aron, R., Galindo, P. and Zagorodnyuk, A., Algebras of symmetric holomorphic functions on ℓp, Bull. Lond. Math. Soc. 35 (2003), 55–64.Google Scholar
2.Aron, R. and Berner, P., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106(1) (1978), 3–24.Google Scholar
3.Aron, R., Cole, B. and Gamelin, T., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93.Google Scholar
4.Aron, R., Falcó, J., García, D. and Maestre, M., Algebras of symmetric holomorphic functions of several complex variables, Rev. Mat. Complut. 31 (2018), 651–672.Google Scholar
5.Aron, R., Galindo, P., García, D. and Maestre, M., Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348(2) (1996), 543–559.Google Scholar
6.Aron, R., Galindo, P., Pinasco, D. and Zalduendo, I., Group-symmetric holomorphic functions on a Banach space, Bull. Lond. Math. Soc. 48(5) (2016), 779–796.Google Scholar
7.Chernega, I., Galindo, P. and Zagorodnyuk, A., Some algebras of symmetric analytic functions and their spectra, Proc. Edinb. Math. Soc. 55 (2012), 125–142.Google Scholar
8.Chernega, I., Galindo, P. and Zagorodnyuk, A., The convolution operation on the spectra of algebras of symmetric analytic functions, J. Math. Anal. Appl. 395 (2012), 569–577.Google Scholar
9.Chernega, I., Galindo, P. and Zagorodnyuk, A., The multiplicative convolution operation on the spectra of algebras of symmetric analytic functions, Rev. Mat. Complut. 27 (2014), 575–585.Google Scholar
10.Deghoul, D., Construction de caractères exceptionnels sur une algèbre de Fréchet, C. R. Acad. Sci. Paris 312, Série I (1991), 579–580.Google Scholar
11.Dineen, S., Complex analysis on infinite dimensional spaces, Springer Monographs in Mathematics (Springer-Verlag, London, 1999).Google Scholar
12.González, M., Gonzalo, R. and Jaramillo, J., Symmetric polynomials on rearrangement invariant function spaces, J. Lond. Math. Soc. 59(2) (1999), 681–697.Google Scholar
13.Nemirovskii, A. S. and Semenov, S. M., On polynomial approximation of functions on Hilbert space, Mat. USSR Sb. 21 (1973), 255–277.Google Scholar
14.Zalduendo, I., A canonical extension for analytic functions on Banach spaces, Trans. Amer. Math. Soc. 320(2) (1990), 747–763.Google Scholar