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Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Author

Listed:
  • Ab Hamid Kawa

    (Department of Mathematics, Maulana Azad National Urdu University, Hyderabad 500032, India
    Department of Mathematics, University Institute of Engineering and Technology, Guru Nanak University, R. R. Dist. Ibrahimpatnam, Hyderabad 501506, India)

  • Turki Alsuraiheed

    (Department of Mathematical Sciences, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • S. N. Hasan

    (Department of Mathematics, Maulana Azad National Urdu University, Hyderabad 500032, India)

  • Shakir Ali

    (Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India)

  • Bilal Ahmad Wani

    (Department of Mathematics, National Institute of Technology, Srinagar 190006, India)

Abstract

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I 1 , and L m : A → A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕ m : A → A and an additive higher map ζ m : A → Z ( A ) , which annihilates every ( n − 1 ) t h commutator p n ( S 1 , S 2 , ⋯ , S n ) with S 1 S 2 = 0 such that L m ( S ) = ϕ m ( S ) + ζ m ( S ) f o r a l l S ∈ A . We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results.

Suggested Citation

  • Ab Hamid Kawa & Turki Alsuraiheed & S. N. Hasan & Shakir Ali & Bilal Ahmad Wani, 2023. "Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions," Mathematics, MDPI, vol. 11(23), pages 1-20, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4770-:d:1287776
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