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Partial isometries in an absolute order unit space


In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.

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

    Book  Google Scholar 

  2. Asimov, L.: Universally well-capped cones. Pacific J. Math. 26, 421–431 (1968)

    MathSciNet  Article  Google Scholar 

  3. Asimov, L.: Complementary cones in dual Banach Spaces. Illinois J. Math. 18, 657–668 (1974)

    MathSciNet  Article  Google Scholar 

  4. Blackadar, B.: K-Theory for Operator Algebras. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  5. Blackadar, B.: Operator Algebras, Theory of \(C^*\)-Algebras and von Neumann Algebras. Springer, Heidelberg New York (2006)

    MATH  Google Scholar 

  6. Bonsall, F.F.: Sublinear functional and ideals in partially ordered vector spaces. Proc. London Math. Soc. 4, 402–418 (1954)

    MathSciNet  Article  Google Scholar 

  7. Bonsall, F.F.: Regular ideals of partially ordered vector spaces. Proc. Lond. Math. Soc. 6, 626–640 (1956)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  9. Edwards, D.A.: On the homeomorphic affine embedding of a locally compact cone into a Banach dual space endowed with the vague topology. Proc. Lond. Math. Soc. 14, 399–414 (1964)

    MathSciNet  Article  Google Scholar 

  10. Ellis, A.J.: The duality of partially ordered normed linear spaces. J. Lond. Math. Soc. 39, 730–744 (1964)

    MathSciNet  Article  Google Scholar 

  11. Gelfand, I., Neumark, M.: On the embedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N. S. 54, 197–213 (1943)

    Google Scholar 

  12. Jameson, G.J.O.: Ordered Linear Spaces, Lecture Notes in mathematics, vol. 141. Springer, Heidelberg New York (1970)

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

    MathSciNet  Article  Google Scholar 

  14. Kadison, R.V.: A representation theory for commutative topological algebras. Mem. Am. Math. Soc. 7, 1 (1951)

    MathSciNet  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  16. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras. Academic Press Inc, London-New York (1983)

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  22. Karn, A.K., Kumar, A.: Isometries of absolute order unit spaces, accepted in Positivity (2019).

  23. Karn, A.K., Vasudevan, R.: Approximate matrix order unit spaces. Yokohama Math. J. 44, 73–91 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Kirchberg, E., Rørdam, M.: Non-simple purely infinite\(\text{ C}^\ast\)-algebras. Am. J. Math. 122, 637–666 (2000)

    MathSciNet  Article  Google Scholar 

  25. Ng, K.F.: The duality of partially ordered Banach spaces. Proc. Lond. Math. Soc. 19, 269–288 (1969)

    MathSciNet  Article  Google Scholar 

  26. Pedersen, G.K.: \(\text{ C}^*\)-Algebras and Their Automorphism Groups. Academic Press Inc, London-New York (1979)

    MATH  Google Scholar 

  27. Rørdam, M.: A simple \(\text{ C}^\ast\)-algebra with a finite and an infinite projection. Acta Math. 191, 109–142 (2003)

    MathSciNet  Article  Google Scholar 

  28. Rørdam, M., Larsen, F., Laustsen, N.J.: An Introduction to K-theory for \(\text{ C}^*\)-Algebras. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  29. Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin-Heidelberg-New York (1974)

    Book  Google Scholar 

  30. Wegge-Olsen, N.E.: K-Theory and \(\text{ C}^*\)-Algebras. Oxford University Press, New York (1993)

    MATH  Google Scholar 

  31. Wong, Y.C., Ng, K.F.: Partially Ordered Topological Vector Spaces. Clarendon Press, Oxford, Oxford Mathematical Monograph (1973)

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We would like to thank the referee for valuable comments and suggestions. The second author was financially supported by the Senior Research Fellowship of the University Grants Commission of India.

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Correspondence to Anil Kumar Karn.

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Communicated by Ngai-Ching Wong.

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Karn, A.K., Kumar, A. Partial isometries in an absolute order unit space. Banach J. Math. Anal. 15, 23 (2021).

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  • Absolutely ordered space
  • Absolute order unit space
  • Absolute matrix order unit space
  • Order projection
  • Partial isometry

Mathematics Subject Classification

  • 46B40
  • 46L05
  • 46L30