ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 64, Issue 1, pp 99–110 | Cite as

Characterization of some derivations on von Neumann algebras via left centralizers

Article

Abstract

Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.

Keywords

Left (right) centralizer Semi Hochschild 2-cocycle Bi-\(\sigma \)-derivaton \(\mathcal {C}\)-derivation von Neumann algebra 

Mathematics Subject Classification

Primary 47B47 Secondary 17B40 46L10 

Notes

Acknowledgements

The author is greatly indebted to the referee for his/her valuable suggestions and careful reading of the paper.

References

  1. 1.
    Ali, S., Haetinger, C.: Jordan \(\alpha \)-centralizers in rings and some applications. Bol. Soc. Paran. Mat. 26, 71–80 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Brešar, M.: Characterizations of derivations on some normed algebras with involution. J. Algebra. 152, 454–462 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brešar, M.: Jordan derivations on semiprime rings. Proc. Am. Math. Soc. 140(4), 1003–1006 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cusack, J.: Jordan derivations on rings. Proc. Am. Math. Soc. 53, 1104–1110 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, New Series, vol. 24. Oxford University Press, New York (2000)MATHGoogle Scholar
  6. 6.
    Dales, G., et al.: Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press, Cambridge (2002)Google Scholar
  7. 7.
    Herstein, I.N.: Jordan derivations of prime rings. Proc. Am. Math. Soc. 8, 1104–1110 (1957)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hosseini, A.: A characterization of derivations on uniformly mean value Banach algebras. Turk. J. Math. 40, 1058–1070 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hosseini, A.: A general characterization of additive maps on semiprime rings. J. Math. Ext. 10, 119–135 (2016)MathSciNetGoogle Scholar
  10. 10.
    Hassani, M., Hosseini, A.: On two variables derivations and generalized centralizers. J. Adv. Res. Pure Math. 6, 38–51 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hou, C., Meng, Q.: Continuity of (\(\alpha, \beta \))-derivations of operator algebras. J. Korean Math. Soc. 48, 823–835 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I-II. Academic Press (1983–1986)Google Scholar
  13. 13.
    Murphy, G.J.: \(C^{\ast }\)-Algebras and Operator Theory. Academic Press, INC., San Diego (1990)MATHGoogle Scholar
  14. 14.
    Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. Lond. Math. Soc. 5, 432–438 (1972)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sait\(\widehat{o}\), K., Maitland Wright, J.D.: On Defining AW*-Algebras and Rickart C*-Algebras. arXiv:1501.02434v1 [math. OA] 11 Jan 2015
  16. 16.
    Takesaki, M.: Theory of Operator Algebras. Springer, Berlin (2001)MATHGoogle Scholar
  17. 17.
    Zalar, B.: On centralizers of semiprime rings. Comment. Math. Univ. Carolin. 32(4), 609–614 (1991)MathSciNetMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2017

Authors and Affiliations

  1. 1.Department of MathematicsKashmar higher education InstituteKashmarIran

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