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Absolutely compatible pair of elements in a von Neumann algebra-II

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas volume 114, Article number: 153 (2020) Cite this article

Abstract

Let A be a unital C\(^*\)-algebra with unity \(1_A\). A pair of elements \(0 \le a, b \le 1_A\) in A is said to be absolutely compatible if, \(\vert a - b \vert + \vert 1_A - a - b \vert = 1_A.\) In this paper we provide a complete description of absolutely compatible pair of strict elements in a von Neumann algebra. The end form of such a pair has a striking resemblance with that of a ‘generic pair’ of projections on a complex Hilbert space introduced by Halmos.

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References

  1. Blackadar, B.: Operator algebras. Springer-Verlag, Heidelberg Berlin (2006)

    Book  Google Scholar 

  2. Halmos, P.R.: Two subspaces. Trans. Am. Math. Soc. 144, 181–189 (1969)

    MathSciNet  Article  Google Scholar 

  3. Jana, N.K., Karn, A.K., Peralta, A.M.: Absolutely compatible pairs in a von Neumann algebra. Electron J Linear Algebra 35, 599–618 (2019)

    MathSciNet  Article  Google Scholar 

  4. Jana, N.K., Karn, A.K., Peralta, A.M.: Contractive linear preservers of absolutely compatible pairs between C\(^*\)-algebras, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RCSM) 113(3), 2731–2741 (2019)

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  7. Pedersen, G.K.: C\(^*\)-algebras and their automorphism groups. Academic Press, London (1979)

    MATH  Google Scholar 

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Acknowledgements

The author is grateful to Antonio M. Peralta for introducing him the notion of ‘strict’ elements. The author is also grateful to the referees for their valuable suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, P.O. Jatni, Khurda, Odisha, 752050, India

    Anil Kumar Karn

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  1. Anil Kumar Karn
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Correspondence to Anil Kumar Karn.

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Karn, A.K. Absolutely compatible pair of elements in a von Neumann algebra-II. RACSAM 114, 153 (2020). https://doi.org/10.1007/s13398-020-00883-7

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  • DOI: https://doi.org/10.1007/s13398-020-00883-7

Keywords

  • von Neumann algebras
  • Absolute compatibility
  • Strict element
  • Generic pair of projections

Mathematics Subject Classification

  • Primary 46L10
  • Secondary 46B40