Ultraweak Continuity of σ-derivations on von Neumann Algebras

  • Madjid Mirzavaziri
  • Mohammad Sal MoslehianEmail author


Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra \(\mathfrak M\). We show that there are a surjective ultraweakly continuous ∗-homomorphism \(\Sigma:\mathfrak M\to\mathfrak M\) and a Σ-derivation \(D:\mathfrak M\to\mathfrak M\) such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ-derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ-derivation on \(\mathfrak M\), then there is an ultraweakly continuous linear mapping \(\Sigma:\mathfrak M\to\mathfrak M\) such that d is a ∗-Σ-derivation.


Derivation ∗-homomorphism ∗-σ-derivation Inner σ-derivation von Neumann algebra Ultraweak topology Weak (operator) topology 

Mathematics Subject Classifications (2000)

Primary 46L57 Secondary 46L05 47B47 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 1. C *- and W *-algebras, symmetry groups, decomposition of states, 2nd edn. Texts and Monographs in Physics. Springer, New York (1987)Google Scholar
  2. 2.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics, 2nd edn. Texts and Monographs in Physics. Springer, Berlin (1997)zbMATHGoogle Scholar
  3. 3.
    Brešar, M., Villena, A.R.: The noncommutative Singer-Wermer conjecture and ϕ-derivations. J. Lond. Math. Soc. (2) 66(3), 710–720 (2002)Google Scholar
  4. 4.
    Hartwig, J., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using σ-derivations. J. Algebra 295(2), 314–361 (2006)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hejazian, S., Janfada, A.R., Mirzavaziri, M., Moslehian, M.S.: Achievement of continuity of (φ, ψ)-derivations without linearity. Bull. Belg. Math. Soc.-Simon Stevn. 14(4), 641–652 (2007)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Mirzavaziri, M., Moslehian, M.S.: σ-derivations in Banach algebras. Bull. Iran. Math. Soc. 32(1), 65–78 (2006)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Mirzavaziri, M., Moslehian, M.S.: Automatic continuity of σ-derivations in C *-algebras. Proc. Am. Math. Soc. 134(11), 3319–3327 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mirzavaziri, M., Moslehian, M.S.: A Kadison-Sakai type theorem. Bull. Aust. Math. Soc. (in press)Google Scholar
  9. 9.
    Sinclair, A.M., Smith, R.R.: Hochschild Cohomology of von Neumann Algebras. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  10. 10.
    Takesaki, M.: Theory of Operator Algebras. I, Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences. 124. Operator Algebras and Non-commutative Geometry, 5. Springer, Berlin (2002)Google Scholar
  11. 11.
    Zhan, J.M., Tan, Z.S.: T-local derivations of von Neumann algebras. Northeast. Math. J. 20(2), 145–152 (2004)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran
  2. 2.Centre of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

Personalised recommendations