JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, I

Title & Authors
JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, I
Kim, Byung-Do;

Abstract
The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $\small{D:A{\rightarrow}A}$ such that $\small{D(x)^3[D(x),x}$$\small{]}$$\small{{\in}rad(A)}$ for all $\small{x{\in}A}$. In this case, we show that $\small{D(A){\subseteq}rad(A)}$.
Keywords
prime and semiprime ring;(Jacobson) radical;Jordan derivation;
Language
English
Cited by
1.
JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, II,;

충청수학회지, 2014. vol.27. 1, pp.65-87
1.
JORDAN DERIVATIONS ON A LIE IDEAL OF A SEMIPRIME RING AND THEIR APPLICATIONS IN BANACH ALGEBRAS, The Pure and Applied Mathematics, 2016, 23, 4, 347
2.
Generalized derivations on Lie ideals in prime rings, Czechoslovak Mathematical Journal, 2015, 65, 1, 179
3.
Engel conditions of generalized derivations on Lie ideals and left sided ideals in prime rings and Banach Algebras, Afrika Matematika, 2016, 27, 7-8, 1391
4.
JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, II, Journal of the Chungcheong Mathematical Society , 2014, 27, 1, 65
References
1.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Berlin-Heidelberg-New York, 1973.

2.
M. Bresar, Derivations of noncommutative Banach algebras. II, Arch. Math. 63 (1994), no. 1, 56-59.

3.
M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1003-1006.

4.
L. O. Chung and J. Luh, Semiprime rings with nilpotent derivatives, Canad. Math. Bull. 24 (1981), no. 4, 415-421.

5.
B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.

6.
B. D. Kim, On the derivations of semiprime rings and noncommutative Banach algebras, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 1, 21-28.

7.
B. D. Kim, Derivations of semiprime rings and noncommutative Banach algebras, Commun. Korean Math. Soc. 17 (2002), no. 4, 607-618.

8.
B. D. Kim, Jordan derivations of semiprime rings and noncommutative Banach algebras. I, J. Korea Soc. Math. Educ. Ser. B. Pure Appl. Math. 15 (2008), no. 2, 179-201.

9.
B. D. Kim, Jordan derivations of semiprime rings and noncommutative Banach algebras. II, J. Korea Soc. Math. Educ. Ser. B. Pure Appl. Math. 15 (2008), no. 3, 259-296.

10.
K. H. Park and B. D. Kim, On continuous linear Jordan derivations of Banach algebras, J. Korea Soc. Math. Educ. Ser. B. Pure Appl. Math. 16 (2009), no. 2, 227-241.

11.
A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209-214.

12.
I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264.

13.
M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), no. 3, 435-460.

14.
J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 83-88.

15.
J. Vukman, On derivations in prime rings and Banach algebras, Proc. Amer. Math. Soc. 116 (1992), no. 4, 877-884.