Darvish, Vahid; Nouri, Mojtaba; Razeghi, Mehran; Taghavi, Ali Nonlinear \(\ast\)-Jordan triple derivation on prime \(\ast\)-algebras. (English) Zbl 1435.16011 Rocky Mt. J. Math. 50, No. 2, 543-549 (2020). Summary: Let \(\mathcal{\mathcal{A}}\) be a prime \(\ast\)-algebra and suppose that \(\Phi\) preserves triple \(\ast\)-Jordan derivation on \(\mathcal{\mathcal{A}}\), that is, for every \(A, B \in \mathcal{\mathcal{A}}\), \[\Phi (A \Diamond B \Diamond C) = \Phi (A) \Diamond B \Diamond C + A \Diamond \Phi (B) \Diamond C + A \Diamond B \Diamond \Phi (C),\] where \(A \Diamond B = A B + B A^\ast \). Then \(\Phi\) is additive. Moreover, if \(\Phi (\alpha I)\) is selfadjoint for \(\alpha \in \{1,i\}\), then \(\Phi\) is a \(\ast\)-derivation. Cited in 1 ReviewCited in 8 Documents MSC: 16W25 Derivations, actions of Lie algebras 46J10 Banach algebras of continuous functions, function algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 47B48 Linear operators on Banach algebras Keywords:\(\ast\)-Jordan triple derivation; derivation; prime algebra × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Z. Bai and S. Du, “The structure of nonlinear Lie derivation on von Neumann algebras”, Linear Algebra Appl. 436:7 (2012), 2701-2708. · Zbl 1272.47046 · doi:10.1016/j.laa.2011.11.009 [2] J. Cui and C.-K. Li, “Maps preserving product \(XY-YX^*\) on factor von Neumann algebras”, Linear Algebra Appl. 431:5-7 (2009), 833-842. · Zbl 1183.47031 · doi:10.1016/j.laa.2009.03.036 [3] C. Li, F. Lu, and X. Fang, “Nonlinear mappings preserving product \(XY+YX^\ast\) on factor von Neumann algebras”, Linear Algebra Appl. 438:5 (2013), 2339-2345. · Zbl 1276.47047 · doi:10.1016/j.laa.2012.10.015 [4] C. Li, F. Lu, and X. Fang, “Non-linear \(\xi \)-Jordan \(\ast \)-derivations on von Neumann algebras”, Linear Multilinear Algebra 62:4 (2014), 466-473. · Zbl 1293.47037 · doi:10.1080/03081087.2013.780603 [5] C. J. Li, F. F. Zhao, and Q. Y. Chen, “Nonlinear skew Lie triple derivations between factors”, Acta Math. Sin. \((\) Engl. Ser.\() 32\):7 (2016), 821-830. · Zbl 1362.47025 · doi:10.1007/s10114-016-5690-1 [6] L. Molnár, “A condition for a subspace of \(\mathcal{B}(H)\) to be an ideal”, Linear Algebra Appl. 235 (1996), 229-234. · Zbl 0852.46021 · doi:10.1016/0024-3795(94)00143-X [7] A. Taghavi, H. Rohi, and V. Darvish, “Non-linear \(*\)-Jordan derivations on von Neumann algebras”, Linear Multilinear Algebra 64:3 (2016), 426-439. · Zbl 1353.46049 [8] A. Taghavi, V. Darvish, and H. Rohi, “Additivity of maps preserving products \(AP\pm PA^*\) on \(C^*\)-algebras”, Math. Slovaca 67:1 (2017), 213-220. · Zbl 1399.47106 [9] A. Taghavi, M. Nouri, M. Razeghi, and V. Darvish, “Non-linear \(\lambda \)-Jordan triple \(\ast \)-derivation on prime \(\ast \)-algebras”, Rocky Mountain J. Math. 48:8 (2018), 2705-2716. · Zbl 1466.46042 · doi:10.1216/RMJ-2018-48-8-2705 [10] A. Taghavi, M. Nouri, M. Razeghi, and V. Darvish, “A note on non-linear \(*\)-Jordan derivations on \(*\)-algebras”, Math. Slovaca 69:3 (2019), 639-646. · Zbl 1466.46040 · doi:10.1515/ms-2017-0253 [11] W. · Zbl 1263.46058 · doi:10.1016/j.laa.2012.05.032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.