• Hou, Chengjun ;
  • Meng, Qing
  • Received : 2010.04.15
  • Published : 2011.07.01


We investigate the continuity of (${\alpha},{\beta}$)-derivations on B(X) or $C^*$-algebras. We give some sufficient conditions on which (${\alpha},{\beta}$)-derivations on B(X) are continuous and show that each (${\alpha},{\beta}$)-derivation from a unital $C^*$-algebra into its a Banach module is continuous when and ${\alpha}$ ${\beta}$ are continuous at zero. As an application, we also study the ultraweak continuity of (${\alpha},{\beta}$)-derivations on von Neumann algebras.


automatic continuity;(${\alpha},{\beta}$)-derivation;Banach algebra;algebra;C*-algebra


  1. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York- Heidelberg, 1973.
  2. E. Christensen, Derivations of nest algebras, Math. Ann. 229 (1977), no. 2, 155-161.
  3. H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, 24. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.
  4. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.
  5. R. V. Kadison, Derivations of operator algebras, Ann. of Math. (2) 83 (1966), 280-293.
  6. R. V. Kadison and J. Ringrose, Fundamentals of the Theory of Operators, Vol 1: Elementary Theory; Vol 2: Advanced Theory, Graduate Studies in Mathematics, Vol. 15, Vol. 16, American Mathematical Society, 1997.
  7. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858.
  8. I. Kaplansky, Derivation of Banach algebras, Seminar on Analytic Functions, Vol. II. Institute for Advanced Study, Prnceton, 1958.
  9. W. S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695-698.
  10. M. Mirzavaziri and M. S. Moslehian, Automatic continuity of $\sigma$-derivations on $C^{{\ast}{\ast}}$- algebras, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3319-3327.
  11. M. Mirzavaziri and M. S. Moslehian, $\sigma$-derivations in Banach algebras, Bull. Iranian Math. Soc. 32 (2006), no. 1, 65-78.
  12. M. S. Moslehian, Hyers-Ulam-Rassias stability of generalized derivations, Int. J. Math. Math. Sci. 2006 (2006), Art. ID 93942, 8 pp.
  13. J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432-438.
  14. S. Sakai, On a conjecture of Kaplansky, Tohoku Math. J. (2) 12 (1960), 31-33.
  15. A. R. Villena, Automatic continuity in associative and nonassociative context, Irish Math. Soc. Bull. No. 46 (2001), 43-76.

Cited by

  1. Jordan (α,β)-Derivations on Operator Algebras vol.2017, 2017,
  2. Characterization of some derivations on von Neumann algebras via left centralizers 2017,
  3. Continuity and Structure of Generalized $$\varvec{(\phi ,\psi )}$$ ( ϕ , ψ ) -Derivations 2017,