• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권2호
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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).

• 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 $x{\in}A$.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.91-102
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• 2009

Using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras and of derivations on Banach algebras for the 3-variable Cauchy functional equation.

• Korean Journal of Mathematics
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• 제22권4호
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• pp.659-670
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• 2014

In this paper, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach algebras and of derivations on fuzzy Banach algebras associated to the Cauchy-Jensen functional equation.

• Kyungpook Mathematical Journal
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• 제53권4호
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• pp.497-505
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• 2013

The noncommutative Singer-Wermer conjecture states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of ${\phi}$-derivations to be nilpotent. Moreover, we show that the spectral boundedness of ${\phi}$-derivations implies that they leave each primitive ideal of Banach algebras invariant.

• 대한수학회논문집
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• 제12권2호
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• pp.251-256
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• 1997

We show that a strong system of derivations ${D_0, D_1,\cdots,D_m}$ on a commutative Banach algebra A is contained in the radical of A if it satisfies one of the following conditions for separating spaces; (1) $\partial(D_i) \subseteq rad(A) and \partial(D_i) \subseteq K D_i(rad(A))$ for all i, where $K D_i(rad(A)) = {x \in rad(A))$ : for each $m \geq 1, D^m_i(x) \in rad(A)}$. (2) $(D^m_i) \subseteq rad(A)$ for all i and m. (3) $\bar{x\partial(D_i)} = \partial(D_i)$ for all i and all nonzero x in rad(A).

• Kyungpook Mathematical Journal
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• 제47권1호
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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.

• 충청수학회지
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• 제18권2호
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• pp.175-193
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• 2005

In [1], the concept of generalized (${\theta}$, ${\phi}$)-derivations on rings was introduced. In this paper, we introduce the concept of generalized (${\theta}$, ${\phi}$)-derivations on Poisson Banach algebras and of generalizd (${\theta}$, ${\phi}$)-derivations on Jordan Banach algebras, and prove the Cauchy-Rassias stability of generalized (${\theta}$, ${\phi}$)-derivations on Poisson Banach algebras and of generalized (${\theta}$, ${\phi}$)-derivations on Jordan Banach algebras.

• 호남수학학술지
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• 제43권1호
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• pp.68-77
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• 2021

This paper considers pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. We characterize pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. Moreover, we define pseudo derivations on the above stated Banach algebras and characterize this new notion on those algebras.

• 충청수학회지
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• 제27권1호
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• pp.65-87
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• 2014

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. We show that if there exists a continuous linear Jordan derivation D : A ${\rightarrow}$ A such that [D(x), x]$D(x)^3{\in}$ rad(A) for all $x{\in}A$, then D(A) ${\subseteq}$ rad(A).