• Communications of the Korean Mathematical Society
• /
• v.17 no.2
• /
• pp.245-252
• /
• 2002

In this paper we obtain some results concerning Jordan derivations and Jordan left derivations mapping into the Jacobson radical. Our main result is the following : Let d be a Jordan derivation (resp. Jordan left derivation) of a complex Banach algebra A. If d$^2$(x) = 0 for all x $\in$ A, then we have d(A) ⊆ red(A)

• Kyungpook Mathematical Journal
• /
• v.54 no.2
• /
• pp.189-195
• /
• 2014

Let X be a compact metric space, B be a unital commutative Banach algebra and ${\alpha}{\in}(0,1]$. In this paper, we first define the vector-valued (B-valued) ${\alpha}$-Lipschitz operator algebra $Lip_{\alpha}$ (X, B) and then study its structure and characterize of its maximal ideal space.

• Journal of the Chungcheong Mathematical Society
• /
• v.11 no.1
• /
• pp.115-121
• /
• 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
• /
• v.11 no.1
• /
• pp.123-128
• /
• 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
• /
• v.13 no.1_2
• /
• pp.425-432
• /
• 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
• /
• v.5 no.2
• /
• pp.539-546
• /
• 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
• /
• v.41 no.4
• /
• pp.697-705
• /
• 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 applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.305-317
• /
• 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
• /
• v.47 no.1
• /
• pp.119-125
• /
• 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.

• Kyungpook Mathematical Journal
• /
• v.53 no.4
• /
• pp.497-505
• /
• 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.