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Science China Mathematics

, Volume 62, Issue 3, pp 553–568 | Cite as

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

  • Antonio M. PeraltaEmail author
  • Ryotaro Tanaka
Articles

Abstract

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank smaller than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : EB. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands (and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley’s problem admits a positive answer.

Keywords

Tingley’s problem extension of isometries JB*-triples compact operators 

MSC(2010)

47B49 46A22 46B20 46B04 46A16 46E40 

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Notes

Acknowledgements

This work was supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (Grant No. MTM2014-58984-P), Junta de Andalucía (Grant No. FQM375), Grants-in-Aid for Scientific Research (Grant No. 16J01162) and Japan Society for the Promotion of Science.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Faculty of MathematicsKyushu UniversityFukuokaJapan

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