• Title/Summary/Keyword: Jordan Derivations

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τ-CENTRALIZERS AND GENERALIZED DERIVATIONS

  • Zhou, Jiren
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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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.

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON C*-ALGEBRAS

  • Taghavi, Ali;Akbari, Aboozar
    • Korean Journal of Mathematics
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    • v.26 no.2
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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.

ON GENERALIZED JORDAN DERIVATIONS OF GENERALIZED MATRIX ALGEBRAS

  • Ashraf, Mohammad;Jabeen, Aisha
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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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, 𝝃MN, ΩNM). 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.

CHARACTERIZATIONS OF (JORDAN) DERIVATIONS ON BANACH ALGEBRAS WITH LOCAL ACTIONS

  • Jiankui Li;Shan Li;Kaijia Luo
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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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.

NOTE OF JORDAN DERIVATIONS ON BANACH ALGEBRAS

  • Chang, Ick-Soon;Kim, Hark-Mahn
    • Journal of applied mathematics & informatics
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    • v.9 no.1
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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.

THE RESULTS CONCERNING JORDAN DERIVATIONS

  • Kim, Byung Do
    • Journal of the Chungcheong Mathematical Society
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    • v.29 no.4
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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 $x{\in}A$.

THE JORDAN DERIVATIONS OF SEMIPRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

  • Kim, Byung-Do
    • Journal of the Chungcheong Mathematical Society
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    • v.29 no.4
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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 $D(A){\subseteq}rad(A)$.