Skip to main content
SpringerLink
Log in

On a characterization of commutativity for \(C^*\)-algebras via gyrogroup operations

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We show that a unital \(C^*\)-algebra is commutative if and only if the gyrogroup of the set of positive invertible elements is in fact a group.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Abe, O. Hatori, Generalized Gyrovector spaces and a Mazur-Ulam theorem. Publ. Math. Debrecen, 87, 393–413 (2015)

  2. R. Beneduci, L. Molnár, On the standard K-loop structures of positive invertible elements in a \(C^*\)-algebra. J. Math. Anal. Appl. 420, 551–562 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.J. Crabb, J. Duncan, C.M. McGregor, Characterizations of commutativity for \(C^*\)-algebras. Glasgow Math. J. 15, 172–175 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-S. Jeang, C.-C. Ko, On the commutativity of \(C^*\)-algebras. Manuscr. Math. 115, 195–198 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Ji, J. Tomiyama, On a characterizations of commutativity of \(C^*\)-algebras. Proc. Am. Math. Soc. 131, 3845–3849 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Kiechle, Theory of K-Loops. Lecture Notes in Mathematics, vol. 1778 (Springer, Berlin, 2002)

  7. L. Molnár, A few conditions for a \(C^*\)-algebra to be commutative, Abstr. Appl. Anal. Article ID 705836 (2014)

  8. T. Ogasawara, A theorem on operator algebras. J. Sci. Hiroshima Univ. 18, 307–309 (1955)

    MathSciNet  MATH  Google Scholar 

  9. L.V. Sabinin, L.L. Sabinina, L.V. Sbitneva, On the notion of gyrogroup. Aequ. Math. 56, 11–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Sherman, Order in operator algebras. Am. J. Math. 73, 227–232 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Uchiyama, Commutativity of selfadjoint operators. Pac. J. Math. 161, 385–392 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. A.A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity (World Scientific, Hackensack, 2008)

    Book  MATH  Google Scholar 

  13. W. Wu, An order characterization of commutativity for \(C^*\)-algebras. Proc. Am. Math. Soc. 129, 983–987 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, Japan

    Toshikazu Abe

  2. Department of Mathematics, Faculty of Science, Niigata University, Niigata, Japan

    Osamu Hatori

Authors
  1. Toshikazu Abe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Osamu Hatori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osamu Hatori.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abe, T., Hatori, O. On a characterization of commutativity for \(C^*\)-algebras via gyrogroup operations. Period Math Hung 72, 248–251 (2016). https://doi.org/10.1007/s10998-016-0126-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-016-0126-3

Keywords

Mathematics Subject Classification

Search

Navigation