Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 67 -- 78

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NON-LINEAR TRIPLE PRODUCT A*B - B*A DERIVATIONS ON *-ALGEBRAS

Mohammad Shavandi and Ali Taghavi

Abstract. Let 𝒜 be a unital prime *-algebra that possesses a nontrivial projection, and let Φ : 𝒜 → 𝒜 be a non-linear map which satisfies

Φ(A ◇ B ◇ C) = Φ(A)◇ B ◇ C + A ◇ Φ(B) ◇ C + A ◇ B ◇ Φ(C)
for all A, B, C∈𝒜, where A ◇ B = A*B - B*A. Then, if Φ(α I⁄2) is self-adjoint map for α∈ {1,i} we show that Φ is additive *-derivation.

2020 Mathematics Subject Classification: 46J10; 47B48; 46L10
Keywords: new product derivation; prime *-algebra; additive map

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References

  1. Z. Bai, S. Du, The structure of non-linear Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 436 (2012), 2701-2708. MR2890027. Zbl 1272.47046.

  2. E. Christensen, Derivations of nest algebras, Ann. Math. 229 (1977), 155-161. MR0448110. Zbl 0356.46057.

  3. J. Cui, C.K. Li, Maps preserving product XY - Y X* on factor von Neumann algebras, Linear Algebra Appl. 431, (2009), 833-842. MR2535555. Zbl 1183.47031.

  4. C. Li, F. Zhao, Q. Chen,Nonlinear maps preserving product X*Y+Y*X on von Neumann algebras, Bulletin of the Iranian Mathematical Society, 44 (2018), 729–738. MR3829633. Zbl 07040690.

  5. R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras I, New York, Academic Press, 1983. MR0719020. Zbl 0888.46039.

  6. R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras II, New York, Academic Press, 1986. MR0859186. Zbl 0991.46031.

  7. C. Li, F. Lu, X. Fang, Nonlinear ξ-Jordan *-derivations on von Neumann algebras, Linear and Multilinear Algebra. 62 (2014), 466-473. MR3177041. Zbl 1293.47037

  8. C. Li, F. Lu, X. Fang, Nonlinear mappings preserving product XY+YX* on factor von Neumann algebras, Linear Algebra Appl. 438 (2013), 2339-2345. MR3005295. Zbl 1276.47047.

  9. C. Li, F Zhao, Q. Chen, Nonlinear Skew Lie Triple Derivations between Factors, Acta Mathematica Sinica, 32 (2016), 821--830. MR3508624. Zbl 1362.47025.

  10. C. R. Miers, Lie homomorphisms of operator algebras, Pacific J Math. 38 (1971), 717--735. MR0308804. Zbl 0204.14803.

  11. L. Molnár, A condition for a subspace of B(H) to be an ideal, Linear Algebra Appl. 235 (1996), 229-234. MR1374262. Zbl 0852.46021.

  12. S. Sakai, Derivations of W*-algebras, Ann. Math. 83 (1966), 273-279. MR0193528. Zbl 0139.30601.

  13. P. Šemrl, Additive derivations of some operator algebras, Illinois J. Math. 35 (1991), 234-240. MR1091440. Zbl 0705.46035

  14. P. Šemrl, Ring derivations on standard operator algebras, J. Funct. Anal. 112 (1993), 318-324. MR1213141. Zbl 0801.47024.

  15. M. Shavandi, A. Taghavi, Maps preserving n-tuple A*B - B*A derivations on factor von Neumann algebras, Publ. de l'Institut Mathematique, Vol. 113, Issue 127, (2023), 131-140. doi.org/10.2298/PIM2327131S. MR4599721. Zbl 7738163.

  16. A. Taghavi, V. Darvish, H. Rohi, Additivity of maps preserving products AP± PA* on C*-algebras, Mathematica Slovaca 67 (2017), 213--220. MR3630166. Zbl 1399.47106.

  17. A. Taghavi, H. Rohi, V. Darvish, Non-linear *-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), 426--439. MR3439441. Zbl 1353.46049.

  18. W. Yu, J. Zhang, Nonlinear *-Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 437 (2012), 1979-1991. MR2950465. Zbl an:1263.46058.

  19. F. Zhang, Nonlinear skew Jordan derivable maps on factor von Neumann algebras, Linear Multilinear Algebra 64 (2016), 2090--2103. MR3521159. Zbl 1353.47073.



Mohammad Shavandi
University of Mazandaran,
Faculty of Mathematical Sciences, Department of Mathematics,
P. O. Box 47416-1468, Babolsar, Iran.
e-mail: mo.shavandi@gmail.com
Ali Taghavi
Faculty of Mathematical Sciences, Department of Mathematics,
P. O. Box 47416-1468, Babolsar, Iran.
e-mail: taghavi@umz.ac.ir

http://www.utgjiu.ro/math/sma