• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.489-499
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• 1998

Let A be the non-commutative Banach algebra with identity satisfying certain conditions. We show that if D is a derivation on A, then D(A) is contained in the radical of A.

• Journal of applied mathematics & informatics
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• v.6 no.2
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• pp.611-630
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• 1999

In this paper we show that the Derivation D(A) on the non-commutative Banach algebra A with identity satisfying certain conditions is contained in the radical of A and will show some examples satisfying such properties.

• Journal of applied mathematics & informatics
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• v.7 no.3
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• pp.995-1004
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• 2000

In this paper we shall give a slight generalization of J. Vukman's Theorem. And show from the result that the image of a continuous linear Jordan derivation on a noncommutative Banach algebra A is contained in the radical under the condition [D(x),x]E(x) ${\in}$ rad(A) for all $x{\in}A$ . And we show some properties of the derivations on noncommutative Banach algebras.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.607-618
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• 2002

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A \longrightarrowA such that D($\chi$)[D($\chi$),$\chi$]D($\chi$) $\in$ rad(A) for all $\chi$ A. In this case, we have D(A) ⊆ rad(A).

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.211-224
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• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.709-718
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• 2001

In this paper we show that if D and G are continuous linear Jordan derivations on a Banach algebra A satisfying [D(x), x]x - x[G(x),x] $\epsilon$ rad(A)for all $\epsilon$ A, then both D and G map A into rad(A).

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.179-201
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D\;:\;A{\rightarrow}A$ such that $D(x)[D(x),x]^2\;{\in}\;rad(A)$ or $[D(x), x]^2 D(x)\;{\in}\;rad(A)$ for all $x\;{\in}\ A$. In this case, we have $D(A)\;{\subseteq}\;rad(A)$.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.425-432
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• 2003

Our main goal is to show that if there exists a continuous linear Jordan derivation D on a noncommutative Banach algebra A such that n$^{x}$ D(x)n+xD(x)x$^{n}$ $\in$ rad(A) for all x $\in$ A, then D maps A into rad(A).

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.367-372
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• 1995

In 1955 Singer and Wermer [12] proved that every bounded derivation on a commutative Banach algebra maps into its radical. They conjectured that the continuity of the derivation in their theorm can be removed. In 1988 Thomas [13] proved their conjecture ; Every derivation on a commutative Banach algebra maps into its radical. For noncommutative versions, in 1984 B. Yood [15] proved that the continuous derivations on Banach algebras satisfing [D(a),b] $\in$ Rad(A) for all a, b $\in$ A have the radical range, where [a,b] will be denote the commutator ab-ba. In 1990 M.Bresar and J.Vukman [1] have generlized Yood's result, that is, the continuous linear Jordan derivation on Banach algebra that satisfies [D(a),a] $\in$ Rad(A) for all a $\in$ A has the radical range. In next year Mathieu and Murphy [5] proved that every bounded centralizing derivation on Banach algebras has its image in the radical. Mathieu and Runde [6] removed the boundedness of that.