Glasnik Matematicki, Vol. 47, No. 1 (2012), 95-104.

ON FUNCTIONAL EQUATIONS RELATED TO DERIVATIONS IN SEMIPRIME RINGS AND STANDARD OPERATOR ALGEBRAS

Nejc Širovnik

Abstract.   In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra. Suppose there exist linear mappings D,G:A(X) → L(X) satisfying the relations D(A³)=D(A²)A+A²G(A),G(A³)=G(A²)A+A²D(A) for all A A(X). In this case there exists B L(X) such that D(A)=G(A)=[A,B] holds for all A A(X).

2010 Mathematics Subject Classification.   16N60, 46B99, 39B42.

Key words and phrases.   Prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation.

DOI: 10.3336/gm.47.1.07

References:

1. K. I. Beidar, M. Brešar, M. A. Chebotar and W. S. Martindale III, On Herstein's Lie map Conjectures. II, J. Algebra 238 (2001), 239-264.
   MathSciNet     CrossRef

2. K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Marcel Dekker, Inc., New York, 1996.
   MathSciNet    

3. M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006.
   MathSciNet     CrossRef

4. M. Brešar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228.
   MathSciNet     CrossRef

5. M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. Vol. 37 (1988), 321-322.
   MathSciNet     CrossRef

6. M. Brešar and J. Vukman, Jordan (θ,φ)-derivations, Glas. Mat. Ser. III 26(46) (1991), 13-17.
   MathSciNet    

7. P. R. Chernoff, Representations, automorphisms and derivations of some Operator Algebras, J. Functional Analysis 12 (1973), 275-289.
   MathSciNet     CrossRef

8. J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.
   MathSciNet     CrossRef

9. I. N. Herstein, Jordan derivations on prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1119.
   MathSciNet     CrossRef

10. I. Kosi-Ulbl and J. Vukman, A note on derivations in semiprime rings, Int. J. Math. Math. Sci. 20 (2005), 3347-3350.
    MathSciNet     CrossRef

11. I. Kosi-Ulbl and J. Vukman, On derivations in rings with involution, Int. Math. J. 6 (2005), 81-91.
    MathSciNet    

12. I. Kosi-Ulbl and J. Vukman, An identity related to derivations of standard operator algebras and semisimple H^*-algebras, Cubo 12 (2010), 95-102.
    MathSciNet     CrossRef

13. P. Šemrl, Ring derivations on standard operator algebras, J. Functional Analysis 112 (1993), 318-324.
    MathSciNet     CrossRef

14. J. Vukman, On automorphisms and derivations of operator algebras, Glas. Mat. Ser. III 19(39) (1984), 135-138.
    MathSciNet    

15. J. Vukman, Identities with derivations and automorphisms on semiprime rings, Int. J. Math. Math. Sci. (2005), 1031-1038.
    MathSciNet     CrossRef

16. J. Vukman, I. Kosi-Ulbl and D. Eremita, On certain equations in rings, Bull. Austral. Math. Soc. Vol. 71 (2005), 53-60.
    MathSciNet     CrossRef

17. J. Vukman, On derivations of algebras with involution, Acta Math. Hungar. 112 (2006), 181-186.
    MathSciNet     CrossRef

18. J. Vukman, On derivations of standard operator algebras and semisimple H^*-algebras, Studia Sci. Math. Hungar. 44 (2007), 57-63.
    MathSciNet     CrossRef

19. J. Vukman, Identities related to derivations and centralizers on standard operator algebras, Taiwanese J. Math. 11 (2007), 255-265.
    MathSciNet    

20. J. Vukman, Some remarks on derivations in semiprime rings and standard operator algebras, Glas. Mat. Ser. III 46(66) (2011), 43-48.
    MathSciNet     CrossRef