Startseite On certain equations in semiprime rings and standard operator algebras
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On certain equations in semiprime rings and standard operator algebras

  • Nadeem ur Rehman ORCID logo EMAIL logo
Veröffentlicht/Copyright: 8. August 2018
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Abstract

The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let (X) be the algebra of all bounded linear operators of X into itself and let 𝒜(X)(X) be a standard operator algebra. Suppose there exist linear mappings ,𝒢:𝒜(𝒳)(𝒳) satisfying the relations

(𝒜m+n)=(𝒜m)𝒜n+𝒜m𝒢(𝒜n),
𝒢(𝒜m+n)=𝒢(𝒜m)𝒜n+𝒜m(𝒜n)

for all 𝒜𝒜(𝒳) and some fixed integers m,n1. Then there exists (𝒳), such that (𝒜)=𝒜-𝒜 for all 𝒜(𝒳), where (𝒳) denotes the ideal of all finite rank operators in (X), and (𝒜m)=𝒜m-𝒜m for all 𝒜𝒜(𝒳).

MSC 2010: 16W25; 16W20; 16N60

Acknowledgements

The author is greatly indebted to the referee for his/her valuable suggestions which have improved the paper immensely.

References

[1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973. 10.1007/978-3-642-65669-9Suche in Google Scholar

[2] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1003–1006. 10.1090/S0002-9939-1988-0929422-1Suche in Google Scholar

[3] M. Brešar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), no. 1, 218–228. 10.1016/0021-8693(89)90285-8Suche in Google Scholar

[4] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Aust. Math. Soc. 37 (1988), no. 3, 321–322. 10.1017/S0004972700026927Suche in Google Scholar

[5] P. R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Funct. Anal. 12 (1973), 275–289. 10.1016/0022-1236(73)90080-3Suche in Google Scholar

[6] J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321–324. 10.1090/S0002-9939-1975-0399182-5Suche in Google Scholar

[7] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110. 10.1090/S0002-9939-1957-0095864-2Suche in Google Scholar

[8] L. Molnár, On centralizers of an H*-algebra, Publ. Math. Debrecen 46 (1995), no. 1–2, 89–95. 10.5486/PMD.1995.1495Suche in Google Scholar

[9] P. Šemrl, Ring derivations on standard operator algebras, J. Funct. Anal. 112 (1993), no. 2, 318–324. 10.1006/jfan.1993.1035Suche in Google Scholar

[10] N. Širovnik, On functional equations related to derivations in semiprime rings and standard operator algebras, Glas. Mat. Ser. III 47(67) (2012), no. 1, 95–104. 10.3336/gm.47.1.07Suche in Google Scholar

[11] N. ur Rehman, N. Širovnik and T. Bano, On certain functional equations on standard operator algebras, Mediterr. J. Math. 14 (2017), no. 1, Article ID 12. 10.1007/s00009-016-0823-4Suche in Google Scholar

[12] J. Vukman, On automorphisms and derivations of operator algebras, Glas. Mat. Ser. III 19(39) (1984), no. 1, 135–138. Suche in Google Scholar

[13] J. Vukman, Identities with derivations and automorphisms on semiprime rings, Int. J. Math. Math. Sci. (2005), no. 7, 1031–1038. 10.1155/IJMMS.2005.1031Suche in Google Scholar

[14] J. Vukman, On derivations of algebras with involution, Acta Math. Hungar. 112 (2006), no. 3, 181–186. 10.1007/s10474-006-0071-3Suche in Google Scholar

[15] J. Vukman, Identities related to derivations and centralizers on standard operator algebras, Taiwanese J. Math. 11 (2007), no. 1, 255–265. 10.11650/twjm/1500404650Suche in Google Scholar

[16] J. Vukman, On derivations of standard operator algebras and semisimple H*-algebras, Studia Sci. Math. Hungar. 44 (2007), no. 1, 57–63. 10.1556/sscmath.2006.1005Suche in Google Scholar

[17] J. Vukman, Some remarks on derivations in semiprime rings and standard operator algebras, Glas. Mat. Ser. III 46(66) (2011), no. 1, 43–48. 10.3336/gm.46.1.07Suche in Google Scholar

Received: 2018-02-21
Revised: 2018-04-19
Accepted: 2018-07-11
Published Online: 2018-08-08
Published in Print: 2019-07-01

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