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Characterization of Hilbert C*-module higher derivations

  • S. Kh. Ekrami ORCID logo EMAIL logo
Published/Copyright: October 28, 2023
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Abstract

Let be a Hilbert C * -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * -module higher derivations { φ n : } n = 0 with φ 0 = I satisfying

φ n ( x , y z ) = i + j + k = n φ i ( x ) , φ j ( y ) φ k ( z ) ( x , y , z , n { 0 } )

and all Hilbert C * -module derivations { ψ n : } n = 1 satisfying

ψ n ( x , y z ) = ψ n ( x ) , y z + x , ψ n ( y ) z + x , y ψ n ( z ) ( x , y , z , n ) ,

and we show that for every Hilbert C * -module higher derivation { φ n } n = 0 on , there exists a unique sequence of Hilbert C * -module derivations { ψ n } n = 1 on such that

ψ n = k = 1 n ( j = 1 k r j = n ( - 1 ) k - 1 r 1 φ r 1 φ r 2 φ r k )

for all positive integers n, where the inner summation is taken over all positive integers r j with j = 1 k r j = n .

MSC 2020: 46L08; 16W25

Dedicated to Professor Madjid Mirzavaziri


References

[1] M. A. Ansari, M. Ashraf and M. Shamim Akhter, Characterization of Lie-type higher derivations of triangular rings, Georgian Math. J. 30 (2023), no. 1, 33–46. 10.1515/gmj-2022-2195Search in Google Scholar

[2] P. E. Bland, Higher derivations on rings and modules, Int. J. Math. Math. Sci. (2005), no. 15, 2373–2387. 10.1155/IJMMS.2005.2373Search in Google Scholar

[3] W. Cortes and C. Haetinger, On Jordan generalized higher derivations in rings, Turkish J. Math. 29 (2005), no. 1, 1–10. Search in Google Scholar

[4] M. B. Ghaemi and B. Alizadeh, Approximately Higher Hilbert C * -modules derivations, Int. J. Nonlinear Anal. Appl. 1 (2002), no. 2, 36–43. Search in Google Scholar

[5] C. Haetinger, Higher derivations on Lie ideals, TEMA Tend. Mat. Apl. Comput 3 (2002), 141–145. 10.5540/tema.2002.03.01.0141Search in Google Scholar

[6] N. P. Jewell, Continuity of module and higher derivations, Pacific J. Math. 68 (1977), no. 1, 91–98. 10.2140/pjm.1977.68.91Search in Google Scholar

[7] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. 10.2307/2372552Search in Google Scholar

[8] M. Mirzavaziri, Characterization of higher derivations on algebras, Comm. Algebra 38 (2010), no. 3, 981–987. 10.1080/00927870902828751Search in Google Scholar

[9] A. Nowicki, Inner derivations of higher orders, Tsukuba J. Math. 8 (1984), no. 2, 219–225. 10.21099/tkbjm/1496160039Search in Google Scholar

[10] A. Roy and R. Sridharan, Higher derivations and central simple algebras, Nagoya Math. J. 32 (1968), 21–30. 10.1017/S002776300002657XSearch in Google Scholar

[11] H. Saidi, A. R. Janfada and M. Mirzavaziri, Kinds of derivations of Hilbert C * -modules and their operator algebras, Miskolc Math. Notes 16 (2015), no. 1, 453–461. 10.18514/MMN.2015.1108Search in Google Scholar

[12] S. Xu and Z. Xiao, Jordan higher derivation revisited, Gulf J. Math. 2 (2014), no. 1, 11–21. 10.56947/gjom.v2i1.176Search in Google Scholar

Received: 2023-05-21
Revised: 2023-06-27
Accepted: 2023-07-24
Published Online: 2023-10-28
Published in Print: 2024-06-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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