• 대한수학회논문집
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• 제34권3호
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• pp.811-818
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• 2019

Let A be a Banach algebra with rad(A). We show that if there exists a continuous linear Jordan derivation D on A, then $$[D(x),x]D(x)^2{\in}rad(A)$$ if and only if $D(x)[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• 대한수학회지
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• 제47권3호
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• pp.523-535
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• 2010

In this paper, we show that Jordan $\tau$-centralizers and local $\tau$-centralizers are $\tau$-centralizers under certain conditions. We also discuss a new type of generalized derivations associated with Hochschild 2-cocycles and introduce a special local generalized derivation associated with Hochschild 2-cocycles. We prove that if $\cal{L}$ is a CDCSL and $\cal{M}$ is a dual normal unital Banach $alg\cal{L}$-bimodule, then every local generalized derivation of above type from $alg\cal{L}$ into $\cal{M}$ is a generalized derivation.

• Korean Journal of Mathematics
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• 제26권2호
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• pp.285-291
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• 2018

Let $\mathcal{A}$ be a unital $C^*$-algebra. It is shown that additive map ${\delta}:{\mathcal{A}}{\rightarrow}{\mathcal{A}}$ which satisfies $${\delta}({\mid}x{\mid}x)={\delta}({\mid}x{\mid})x+{\mid}x{\mid}{\delta}(x),\;{\forall}x{{\in}}{\mathcal{A}}_N$$ is a Jordan derivation on $\mathcal{A}$. Here, $\mathcal{A}_N$ is the set of all normal elements in $\mathcal{A}$. Furthermore, if $\mathcal{A}$ is a semiprime $C^*$-algebra then ${\delta}$ is a derivation.

• 대한수학회논문집
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• 제35권3호
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• pp.733-744
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• 2020

Let 𝕽 be a commutative ring with unity, A and B be 𝕽-algebras, M be a (A, B)-bimodule and N be a (B, A)-bimodule. The 𝕽-algebra 𝕾 = 𝕾(A, M, N, B) is a generalized matrix algebra defined by the Morita context (A, B, M, N, 𝝃_MN, Ω_NM). In this article, we study generalized derivation and generalized Jordan derivation on generalized matrix algebras and prove that every generalized Jordan derivation can be written as the sum of a generalized derivation and antiderivation with some limitations. Also, we show that every generalized Jordan derivation is a generalized derivation on trivial generalized matrix algebra over a field.

• 대한수학회논문집
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• 제38권2호
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• pp.469-485
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• 2023

Let 𝓐 be a unital Banach *-algebra and 𝓜 be a unital *-𝓐-bimodule. If W is a left separating point of 𝓜, we show that every *-derivable mapping at W is a Jordan derivation, and every *-left derivable mapping at W is a Jordan left derivation under the condition W𝓐 = 𝓐W. Moreover we give a complete description of linear mappings 𝛿 and 𝜏 from 𝓐 into 𝓜 satisfying 𝛿(A)B^* + A𝜏(B)^* = 0 for any A, B ∈ 𝓐 with AB^* = 0 or 𝛿(A)◦B^* + A◦𝜏(B)^* = 0 for any A, B ∈ 𝓐 with A◦B^* = 0, where A◦B = AB + BA is the Jordan product.

• 대한수학회논문집
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• 제17권4호
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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).

• 대한수학회보
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• 제39권2호
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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).

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.381-387
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• 2002

Our main goal is to show that if there Jordan derivation D, G on a noncommutative (n+1)!-torsion free prime ring R such that D($\chi$)$\chi$$^n$+$\chi$$^n$G($\chi$) $\in$ C(R) for all $\chi$ $\in$ R, then we have D=0 and G=0.

• 충청수학회지
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• 제29권4호
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• pp.523-530
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• 2016

Let R be a 3!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. In this case, we show that [D(x), x]D(x) = 0 if and only if D(x)[D(x), x] = 0 for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A). If D is a continuous linear Jordan derivation on A, then we see that $[D(x),x]D(x){\in}rad(A)$ if and only if $[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• 충청수학회지
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• 제29권4호
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• pp.531-542
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• 2016

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that [[D(x),x], x]D(x) = 0 or D(x)[[D(x), x], x] = 0 for all $x{\in}R$. In this case we have $[D(x),x]^3=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $[[D(x),x],x]D(x){\in}rad(A)$ or $D(x)[[D(x),x],x]{\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.