Abstract
Let \({\mathfrak {R}}\) be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent p. In this article, we characterize the Lie-type derivation at zero product as well as idempotent product via certain assumptions. Further, we discuss the characterization of Lie-type derivations by local actions on some operator algebras.
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References
Abdullaev, I.Z.: \(n\)-Lie derivations on von Neumann algebra. Uzbek. Mat. Zh. 5(6), 3–9 (1992)
Ashraf, M., Jabeen, A.: Characterization of Lie type derivation on von Neumann algebras. Bull. Korean Math. Soc. 58(5), 1193–1208 (2021)
Beidar, K.I., Martindale, W.S., Mikhalev, A.V.: Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 196. Marcel Dekker, CRC Press, New York (1996)
Benkovič, D.: Lie triple derivations of unital algebras with idempotents. Linear Multilinear Algebr. 63, 141–165 (2015)
Brešar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents. Proc. Roy. Soc. Edinburgh Sect. A 137, 9–21 (2007)
Halmos, P.R.: A Hilbert Space Problem Book, 2 ed., Graduate Texts in Mathematics, Springer-Verlag New York, (1982)
Ji, P., Qi, W.: Characterizations of Lie derivations of triangular algebras. Linear Algebra Appl. 435(5), 1137–1146 (2011)
Ji, P., Qi, W., Sun, X.: Characterizations of Lie derivations of factor von Neumann algebras. Linear Multilinear Algebr. 61(3), 417–428 (2013)
**g, W., Lu, F.: Lie derivable map**s on prime rings. Linear Multilinear Algebr. 60(2), 167–180 (2012)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I, 1986th edn. Academic Press, New York, II (1983)
Li, C., Fang, X., Lu, F., Wang, T.: Lie triple derivable map**s on rings. Comm. Algebr. 42(6), 2510–2527 (2014)
Lu, F.Y., **g, W.: Characterizations of Lie derivations of \({B(X)}\). Linear Algebr. Appl. 432(1), 89–99 (2010)
Qi, X., Hou, J.: Characterization of Lie derivations on prime rings. Comm. Algebr. 39(10), 3824–3835 (2011)
Qi, X., Hou, J.: Characterization of Lie derivations on von Neumann algebras. Linear Algebr. Appl. 438(1), 533–548 (2013)
Šemrl, P.: Additive derivations of some operator algebras. Illinois J. Math. 35(2), 234–240 (1991)
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Communicated by Miin Huey Ang.
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Jabeen, A. On Lie-Type Derivations of Rings and Standard Operator Algebras by Local Actions. Bull. Malays. Math. Sci. Soc. 46, 51 (2023). https://doi.org/10.1007/s40840-022-01400-2
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DOI: https://doi.org/10.1007/s40840-022-01400-2