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Isometries of absolute order unit spaces

Abstract

We prove that a unital, bijective linear map between absolute order unit spaces is an isometry if and only if it is absolute value preserving. We deduce that, on (unital) JB-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that a unital, bijective \(*\)-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) \(\hbox {C}^*\)-algebras such maps are precisely \(\hbox {C}^*\)-algebra isomorphisms.

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Acknowledgements

The authors are grateful to the referee(s) for their valuable suggestions.

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Correspondence to Anil Kumar Karn.

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The Amit Kumar was financially supported by the Senior Research Fellowship of the University Grants Commission of India.

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Karn, A.K., Kumar, A. Isometries of absolute order unit spaces. Positivity 24, 1263–1277 (2020). https://doi.org/10.1007/s11117-019-00731-y

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  • DOI: https://doi.org/10.1007/s11117-019-00731-y

Keywords

  • Absolutely ordered space
  • Absolute oder unit space
  • Isometry
  • Absolute value preserving maps
  • Absolute matrix order unit space

Mathematics Subject Classification

  • Primary 46B40
  • Secondary 46L05
  • 46L30