Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

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)

    Article  MathSciNet  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)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

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.

Author information

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

Authors
  1. Anil Kumar Karn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anil Kumar Karn.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

Mathematics Subject Classification

Search

Navigation