The Mazur–Ulam property for the space of complex null sequences

ABSTRACT

Given an infinite set $\mathrm{\Gamma }$, we prove that the space of complex null sequences, $c_0(\mathrm{\Gamma })$, satisfies the Mazur–Ulam property, that is, for each Banach space X, every surjective isometry from the unit sphere of $c_0(\mathrm{\Gamma })$ onto the unit sphere of X admits a (unique) extension to a surjective real linear isometry from $c_0(\mathrm{\Gamma })$ to X. We also prove that the same conclusion holds for the finite-dimensional space ${\ell }_{\infty }^m$.

KEYWORDS:

• Tingley’s problem
• Mazur–Ulam property
• extension of isometries

AMS SUBJECT CLASSIFICATIONS:

• Primary: 46B20
• 46A22
• 46B04
• 46B25

Acknowledgements

Most of the results presented in this note were obtained during a visit of A.M. Peralta at Universidad de Almería in July 2017. He would like to thank his coauthors and the Department of Mathematics for their hospitality during his stay. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the final form of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Spanish Ministry of Economy and Competitiveness (MINECO) and European Regional Development Fund [project number MTM2014-58984-P]; Junta de Andalucía [grant number FQM194], [grant number FQM375].