Jordan Higher Derivations on Prime Hilbert C$^*$-modules
Download PDF
Authors: S. K. EKRAMI
DOI: 10.46793/KgJMat2605.817E
Abstract:
Let ℳ be a Hilbert C∗-module. A sequence of linear mappings {φn : ℳ→ℳ}n=0+∞ with φ0 = I, is said to be a Hilbert C∗-module Jordan higher derivation on ℳ, if it satisfies the equation
 |
for all a,b ∈ℳ and each non-negative integer n. In this paper, we show that, if ℳ is prime, then every Hilbert C∗-module Jordan higher derivation {φn}n=0+∞ on ℳ, is a Hilbert C∗-module higher derivation on ℳ. As a consequence, we show that every Hilbert C∗-module Jordan derivation on ℳ, is a Hilbert C∗-module derivation on ℳ.
Keywords:
Derivation, Jordan derivation, higher derivation, Jordan higher derivation, Hilbert C∗-module.
References:
[1] M. J. Atteya, C. Haetinger and D. I. Rasen, (σ,τ)-derivations of semiprime rings, Kragujevac J. Math. 43(2) (2019), 239–246.
[2] M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003–1006.
[3] W. Cortes and C. Haetinger, On Jordan generalized higher derivations in rings, Turk. J. Math. 29 (2005), 1–10.
[4] S. Kh. Ekrami, A note on characterization of higher derivations and their product, J. Mahani Math. Res. 13(1) (2024), 403–415. https://doi.org/10.22103/jmmr.2023.21376.1432
[5] S. Kh. Ekrami, Approximate orthogonally higher ring derivations, Control Optim. App. Math. 7(1) (2022), 93–106. https://doi.org/10.30473/coam.2021.59727.1166
[6] S. Kh. Ekrami, Characterization of Hilbert C∗-module higher derivations, Georgian Math. J. (2023). https://doi.org/10.1515/gmj-2023-2085
[7] S. Kh. Ekrami, Jordan higher derivations, a new approach, J. Algebr. Syst. 10(1) (2022), 167–177. https://doi.org/10.22044/JAS.2021.10636.1527
[8] M. Ferrero and C. Haetinger, Higher derivations and a theorem by Herstein, Quaest. Math. 25(2) (2002), 249–257.
[9] C. Haetinger, Derivações de ordem superior em anéis primos e semiprimos, Ph.D. Thesis, UFRGS, Porto Alegre, Brazil, (2000).
[10] C. Haetinger, Higher derivations on Lie ideals, Trends Comp. App. Math. 3 (2002), 141–145.
[11] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.
[12] N. P. Jewell, Continuity of module and higher derivations, Pacific J. Math. 68 (1977), 91–98.
[13] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858.
[14] M. Mirzavaziri, Characterization of higher derivations on algebras, Comm. Algebra 38 (2010), 981–987.
[15] A. Roy and R. Sridharan, Higher derivations and central simple algebras, Nagoya Math. J. 32 (1968), 21–30.
[16] H. Saidi, A. R. Janfada and M. Mirzavaziri, Kinds of derivations on Hilbert C∗-modules and their operator algebras, Miskolc Math. Notes. 16(1) (2015), 453–461.
[17] E. Tafazzoli and M. Mirzavaziri, Inner higher derivations on algebras, Kragujevac J. Math. 44(2) (2019), 267–273.
[18] S. Xu and Z. Xiao, Jordan higher derivation revisited, Gulf J. Math. 2(1) (2014), 11–21.
Login:
