• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.115-121
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• 1998

The purpose of this paper is to prove the following result: Let A be a noncom mutative Banach algebra. Suppose that there exist continuous linear Jordan derivations $D:A{\rightarrow}A$, $G:A{\rightarrow}A$ such that [$D^2(x)+G(x)$, $x^n$] lies in the Jacobson radical of A for all $x{\in}A$. Then $D(A){\subset}rad(A)$ and $G(A){\subset}rad(A)$.

• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.123-128
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• 1998

The purpose of this paper is to prove the following result: Let A be a noncom mutative Banach algebra. Suppose that $D:A{\rightarrow}A$ is a continuous linear Jordan derivation such that $D^2(x)D(x)^2{\in}rad(A)$ for all $x{\in}A$. Then D maps A into its radical.

• 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).

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.539-546
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• 1998

Let A be a noncommutative Banach algebra. Suppose that a continuos linear Jordan derivation D:A$\longrightarrow$A is such that either $[D^2(\chi),\chi^2]\;or\;(D^2(\chi),\chi]+(D(\chi))^2$ lies in the jacobson radical of A for all $\chi$$\in$A. Then D(A) is contained in the Jacobson radical of A.

• Honam Mathematical Journal
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• v.41 no.4
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• pp.697-705
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• 2019

The article studies the concept of a (𝜑, 𝜓)-biflat and (𝜑, 𝜓)-amenable Banach algebra A, where 𝜑 is a continuous homomorphism on A and 𝜓 ∈ Φ_A. We show if A has a (𝜑, 𝜓)-virtual diagonal, then A is (𝜑, 𝜓)- biflat. In the case where 𝜑(A) is commutative we prove that (𝜑, 𝜓)- biflatness of A implies that A has a (𝜑, 𝜓)-virtual diagonal.

• Journal of the Chungcheong Mathematical Society
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• v.2 no.1
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• pp.81-90
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• 1989

Let $D:C^n(I){\longrightarrow}M$ be a derivation from the Banach algebra of n times continuously differentiable functions on an interval I into a Banach $C^n(I)$-module M. If D is continuous and D(z) is contained in the k-differential subspace, the image of D is contained in the k-differential subspace. The question of when the image of a derivation is contained in the k-differential subspace is discussed.

• The Journal of Natural Sciences
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• v.9 no.1
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• pp.61-68
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• 1997

We shall extend the concept of the sequential Feynman integral over the Yeh-Winner space. We shall show that prove the existence of the sequential Yeh-Feynman integral for all element of Banach algebras.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.305-317
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• 2000

It is well-known that every derivation on a commutative Banach algebra maps into its radical. In this paper we shall give the various algebraic conditions on the ring that every Jordan derivation on a noncommutative ring with suitable characteristic conditions is zero and using this result, we show that every continuous linear Jordan derivation on a noncommutative Banach algebra maps into its radical under the suitable conditions.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.119-125
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• 2007

In this paper, we investigate the following: Let A be a semisimple Banach algebra. Suppose that there exists a linear derivation $f:A{\rightarrow}A$ such that the functional equation $<f(x),x>^2=0$ holds for all $x{\in}A$. Then we have $f=0$ on A.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.227-241
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• 2009

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