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

Ab Hamid Kawa, Turki Alsuraiheed, S. N. Hasan, Shakir Ali () and Bilal Ahmad Wani
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Ab Hamid Kawa: Department of Mathematics, Maulana Azad National Urdu University, Hyderabad 500032, 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

Mathematics, 2023, vol. 11, issue 23, 1-20

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.

Keywords: Lie derivation; multiplicative Lie-type derivation; multiplicative Lie-type higher derivation; von Neumann algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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