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Revista Matemática Complutense

, Volume 32, Issue 1, pp 99–114 | Cite as

Extending surjective isometries defined on the unit sphere of \(\ell _\infty (\Gamma )\)

  • Antonio M. PeraltaEmail author
Article

Abstract

Let \(\Gamma \) be an infinite set equipped with the discrete topology. We prove that the space \(\ell _{\infty }(\Gamma ,{\mathbb {C}}),\) of all complex-valued bounded functions on \(\Gamma \), satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of \(\ell _{\infty }(\Gamma , {\mathbb {C}})\) onto the unit sphere of an arbitrary complex Banach space X admits a unique extension to a surjective real linear isometry from \(\ell _{\infty }(\Gamma ,{\mathbb {C}})\) to X.

Keywords

Tingley’s problem Mazur–Ulam property Extension of isometries \(\ell _{\infty }(\Gamma )\) 

Mathematics Subject Classification

Primary 47B49 Secondary 46A22 46B20 46B04 46A16 46E40 

Notes

Acknowledgements

Author partially supported by the Spanish Ministry of Economy and Competitiveness (MINECO) and European Regional Development Fund Project No. MTM2014-58984-P and Junta de Andalucía Grant FQM375. The author is indebted to the referees for a constructive and thoroughful reports on the first version of this paper. Their valuable suggestions improved, without any doubt, the final presentation of this note.

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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