Skip to main content
SpringerLink
Log in

Examples and Applications of Generalized Gyrovector Spaces

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper we exhibit examples of generalized gyrovector spaces (GGVs) of the sets of positive invertible elements in unital C*-algebras. Then we employ a geometric inequality on the positive cones in these unital C*-algebras. By applying a Mazur–Ulam theorem for GGVs, we analyze Thompson-like isometries on GGVs of positive invertible elements.

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. Abe, T.: The equivalence of gyrocommutative gyrogroups and K-loops (preprint)

  2. Abe T., Hatori O.: Generalized gyrovector spaces and a Mazur–Ulam theorem. Publ. Math. Debrecen 87, 393–413 (2015)

    Article  MathSciNet  Google Scholar 

  3. Abe T., Hatori O.: On a characterization of commutativity for C*-algebras via gyrogroup operations. Period. Math. Hungar. 72, 248–251 (2016)

    Article  MathSciNet  Google Scholar 

  4. Aupetit B.: A Primer on spectral theory. Springer-Verlag, New York (1991)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan J.-T., Li C.-K., Tu C.C.N.: A class of unitarily invariant norms on B(H). Proc. Amer. Math. Soc. 129, 1065–1076 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fletcher, P.T., Joshi, S.: Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors. In: Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis. Lect. Notes Comput. Sci. 3117, 87–98 (2004)

  8. Fujii, J.I.: Differential geometries and their metrics on the positive operators, RIMS Kôkyuroku No. 1632, 28–37. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1632-04. (2009)

  9. Hatori O., Molnár L.: Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C*-algebras. J. Math. Anal. Appl. 409, 158–167 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hiai F., Kosaki H.: Comparison of various means for operators. Jour. Funct. Anal. 163, 300–323 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Honma S., Nogawa T.: Isometries of the geodesic distances for the space of invertible positive operators and matrices. Linear Algebra Appl. 444, 152–164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kiechle, H.: Theorey of K-loops. Lecture Notes in Mathematics, vol. 1778. Springer, Berlin, Heidelberg (2002)

  13. Li C.-K., Tsing N.-K.: Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values. Linear Algebra Appl. 143, 67–97 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moakher M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26, 735–747 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Molnár L.: Thompson isometries of the space of invertible positive operators. Proc. Amer. Math. Soc. 137, 3849–3859 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Molnár L., Nagy G.: Thompson isometries on positive operators: the 2-dimensional case. Electron. J. Linear Algebra 20, 79–89 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Molnár L.: Jordan triple endomorphisms and isometries of spaces of positive definite matrices. Linear Multilinear Algebra 63, 12–33 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Molnár, L.: General Mazur–Ulam type theorems and some applications. In: Arendt, W., Chill, R., Tomilov Y. (eds.) Operator semigroups meet complex analysis, harmonic analysis and mathematical physics, operator theory: advances and applications, vol. 250 (to appear)

  19. Molnár L., Nagy G.: Thompson isometries on positive operators: the 2-dimensional case. Electron. J. Linear Algebra 20, 79–89 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Molnár L., Szokol P.: Transformations on positive definite matrices preserving generalized distance measures. Linear Algebra Appl. 466, 141–159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Sra, S.: Positive definite matrices and the symmetric S-divergence, arXiv:1110.1773v4 [math.FA]

  23. Ungar, A.A.: Analytic hyperbolic geometry and Albert Einstein’s special theory of relativity, World Sci (2008)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Niigata University, Niigata City, Japan

    Osamu Hatori

Authors
  1. Osamu Hatori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osamu Hatori.

Additional information

Dedicated to Thomas Tonev

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hatori, O. Examples and Applications of Generalized Gyrovector Spaces. Results Math 71, 295–317 (2017). https://doi.org/10.1007/s00025-016-0576-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-016-0576-1

Mathematics Subject Classification

Keywords

Search

Navigation