Advertisement

Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 5, pp 1573–1583 | Cite as

Linear Maps on Standard Operator Algebras Characterized by Action on Zero Products

  • Amin Barari
  • Behrooz Fadaee
  • Hoger GhahramaniEmail author
Original Paper
  • 36 Downloads

Abstract

Let \({\mathcal {A}}\) be a unital standard algebra on a complex Banach space \({\mathcal {X}}\) with dim\({\mathcal {X}}\ge 2\). The main result of this paper is to characterize the linear maps \(\delta , \tau : {\mathcal {A}}\rightarrow B({\mathcal {X}})\) satisfying \( A \tau ( B) + \delta ( A) B = 0\) whenever \(A,B\in {\mathcal {A}}\) are such that \(AB=0\). As application of our main result, we determine the linear map \(\delta : {\mathcal {A}}\rightarrow B({\mathcal {H}})\) that has one of the following properties for \(A,B\in {\mathcal {A}}\): if \(AB^{\star }=0\), then \(A\delta (B)^{\star }+\delta (A)B^{\star }=0\), or if \(A^{\star }B=0\), then \( A^{\star }\delta (B)+\delta (A)^{\star }B=0 \), where \({\mathcal {A}}\) is a unital standard operator algebras on a Hilbert space \({\mathcal {H}}\) such that \( {\mathcal {A}} \) is closed under the adjoint operation. We also provide other applications of the main result.

Keywords

Standard operator algebra Linear map Zero product 

Mathematics Subject Classification

47L10 47B49 47B47 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the referee(s) for this paper.

References

  1. 1.
    Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.: Maps preserving zero products. Stud. Math. 193, 131–159 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benkovič, D., Grašič, M.: Generalized derivations on unital algebras determined by action on zero products. Linear Algebra Appl. 445, 347–368 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brešar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents. Proc. R. Soc. Edinb. Sect. A. 137, 9–21 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burgos, M., Ortega, J.S.: On mappings preserving zero products. Linear Multilinear Algebra 61, 323–335 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chebotar, M.A., Ke, W.-F., Lee, P.-H.: Maps characterized by action on zero products. Pac. J. Math. 216, 217–228 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, H.-Y., Liu, K.-S., Mozumder, M.R.: Maps acting on some zero products. Taiwan. J. Math. 18, 257–264 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs. Oxford University Press, Oxford (2000)Google Scholar
  8. 8.
    Ghahramani, H.: On rings determined by zero products. J. Algebra Appl. 12, 1–15 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghahramani, H.: Additive maps on some operator algebras behaving like \((\alpha,\beta )\)-derivations or generalized \((\alpha,\beta )\)-derivations at zero-product elements. Acta Math. Sci. 34B(4), 1287–1300 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ghahramani, H.: On derivations and Jordan derivations through zero products. Oper. Matrices 4, 759–771 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ghahramani, H.: Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal elements. Results Math. 73, 133 (2018).  https://doi.org/10.1007/s00025-018-0898-2 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jing, W., Lu, S., Li, P.: Characterization of derivation on some operator algebras. Bull. Austr. Math. Soc. 66, 227–232 (2002)CrossRefGoogle Scholar
  13. 13.
    Lee, T.-K.: Generalized skew derivations characterized by acting on zero products. Pac. J. Math. 216, 293–301 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, J., Pan, Z.: Annihilator-preserving maps, multipliers, and derivations. Linear Algebra. Appl. 432, 5–13 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Qi, X.F.: Characterization of centralizers on rings and operator algebras. Acta Math. Sin. Chin. Ser. 56, 459–468 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xu, W.S., An, R.L., Hou, J.C.: Equivalent characterization of centralizers on \(B({\cal{H}})\). Acta Math. Sin. English Ser. 32, 1113–1120 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhu, J., Xiong, C.P.: Generalized derivable mappings at zero point on some reflexive operator algebras. Linear Algebra Appl. 397, 367–379 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsPayam Noor UniversityTehranIran
  2. 2.Department of MathematicsUniversity of KurdistanSanandajIran

Personalised recommendations