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Isometries of absolute order unit spaces


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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  1. Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, Heidelberg (1971)

    Book  Google Scholar 

  2. Blecher, D.P., Hay, D.M.: Complete isometries into \(\text{C}^*\)-algebras, ArXiv Priprint (2002), (The main result is featured in the book “Isometries on Banach spaces: function spaces” by R. J. Fleming and J. E. Jamison, Chapman and Hall/CRC, Published September 5, 2019, 208 Pages)

  3. Blecher, D.P., Labuschagne, L.E.: Logmodularity and isometries of operator algebras. Trans. Am. Math. Soc. 355, 1621–1646 (2002)

    MathSciNet  Article  Google Scholar 

  4. Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)

    MathSciNet  Article  Google Scholar 

  5. Chu, C.-H., Wong, N.-C.: Isometries between \(\text{ C }^*\)-algebras. Rev. Mat. Iberoamericana 20, 156–209 (2004)

    MathSciNet  Google Scholar 

  6. Gelfand, I.M., Naimark, M.A.: On the embedding of normed rings into the ring of operators in Hilbert space. Mat. Sb. 12, 87–105 (1943)

    Google Scholar 

  7. Gardener, T.: Linear maps of \(\text{ C }^*\)-algebras preserving the absolute value. Proc. Am. Math. Soc. 76, 271–278 (1979)

    MathSciNet  Google Scholar 

  8. Jana, N.K., Karn, A.K., Peralta, A.M.: Contractive linear preservers of absolutely compatible pairs between \(\text{ C }^*\)-algebras. RACSAM 113(3), 2731–2744 (2019).

    MathSciNet  Article  MATH  Google Scholar 

  9. Jana, N.K., Karn, A.K., Peralta, A.M.: Absolutely compatible pairs in a von Neumann algebra, (Communicated for publication). (arxiv:1801.01216)

  10. Kadison, R.V.: Order properties of self-adjoint operators. Trans. Am. Math. Soc 2, 505–510 (1951)

    MathSciNet  MATH  Google Scholar 

  11. Kadison, R.V.: Isometries of operator algebras. Ann. Math. 54, 325–338 (1951)

    MathSciNet  Article  Google Scholar 

  12. Kakutani, S.: Concrete representation of abstract (\(M\))-spaces. Ann. Math. 42, 994–1024 (1941)

    MathSciNet  Article  Google Scholar 

  13. Karn, A.K.: Orthogonality in \(l_p\)-spaces and its bearing on ordered Banach spaces. Positivity 18(02), 223–234 (2014)

    MathSciNet  Article  Google Scholar 

  14. Karn, A.K.: A p-theory of ordered normed spaces. Positivity 14, 441–458 (2010)

    MathSciNet  Article  Google Scholar 

  15. Karn, A.K.: Orthogonality in \(\text{ C }^*\)-algebras. Positivity 20(03), 607–620 (2016)

    MathSciNet  Article  Google Scholar 

  16. Karn, A.K.: Algebraic orthogonality and commuting projections in operator algebras. Acta Sci. Math. (Szeged) 84, 323–353 (2018)

    MathSciNet  Article  Google Scholar 

  17. Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 138, 503–529 (1983)

    MathSciNet  Article  Google Scholar 

  18. Maitland Wright, J.D., Youngson, M.A.: On isometries of Jordan algebras. J. Lond. Math. Soc. (2) 17, 339–344 (1978)

    MathSciNet  Article  Google Scholar 

  19. Pedersen, G.K.: \(\text{ C }^*\)-algebras and their Automorphism Groups. Academic Press, London (1979)

    MATH  Google Scholar 

  20. Radjabalipour, M., Siddighi, K., Taghavi, Y.: Additive mappings on operator algebras preserving absolute value. Linear Algebra Appl. 327, 197–206 (2001)

    MathSciNet  Article  Google Scholar 

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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).

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  • 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