• 대한수학회보
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• 제61권3호
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• pp.813-823
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• 2024

In this paper, we prove that every 2-local derivation on several classes of C^*-algebras, such as unital properly infinite, type I or residually finite-dimensional C^*-algebras, is a derivation. We show that the following statements are equivalent: (1) every 2-local derivation on a C^*-algebra is a derivation, (2) every 2-local derivation on a unital primitive antiliminal and no properly infinite C^*-algebra is a derivation. We also show that every 2-local derivation on a group C^*-algebra C^*(𝔽) or a unital simple infinite-dimensional quasidiagonal C^*-algebra, which is stable finite antiliminal C^*-algebra, is a derivation.

• 충청수학회지
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• 제19권2호
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• pp.159-175
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• 2006

This paper is a survey on the Hyers-Ulam-Rassias stability of the Jensen functional equation in $C^*$-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Its content is divided into the following sections: 1. Introduction and preliminaries. 2. Approximate isomorphisms in $C^*$-algebras. 3. Approximate isomorphisms in Lie $C^*$-algebras. 4. Approximate isomorphisms in $JC^*$-algebras. 5. Stability of derivations on a $C^*$-algebra. 6. Stability of derivations on a Lie $C^*$-algebra. 7. Stability of derivations on a $JC^*$-algebra.

• Korean Journal of Mathematics
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• 제22권1호
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• 대한수학회논문집
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• 제9권2호
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• pp.299-302
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• 1994

Throughout this paper, let X be a compact Hausdorff space, and let C(X) (resp. $C_{R}$ /(X)) be the complex (resp. real) Banach algebra of all continuous complex-valued (resp. real-valued) functions on X with the pointwise operations and the supremum norm x. A Banach function algebra on X is a Banach algebra lying in C(X) which separates the points of X and contains the constants. A Banach function algebra on X equipped with the supremum norm is called a uniform algebra on X, that is, a uniformly closed subalgebra of C(X) which separates the points of X and contains the constants.(omitted)

• 대한수학회보
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• 제30권1호
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• pp.125-134
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• 1993

Let X be a compact Hausdorff space, and let C(X) (resp. $C_{R}$(X)) be the complex (resp. real) Banach algebra of all continuous complex-valued(resp. real-valued) functions on X with the pointwise operations and the supremum norm x. A Banach function algebra on X is a Banach algebra lying in C(X) which separates the points of X and contains the constants. A Banach function algebra on X equipped with the supremum norm is called a uniform algebra on X, that is, a uniformly closed subalgebra of C(X) which separates the points of X and contains the constants.s.

• 대한수학회보
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• 제43권2호
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• pp.333-341
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• 2006

We analyze a detailed picture of the algebraic structure of $C^*$-algebras generated by isometric representations of the non-amenable semigroup P = {0,2,3,...,N,...}.

• 대한수학회논문집
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• 제14권2호
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• pp.365-371
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• 1999

In this paper, we prove that the range of homomorphism from a C\ulcorner-algebra A into a commutative Banach algebra B whose radical is nil contains no non-zero element of the radical of B. Using this result we show that there is no non-zero homomorphism from a C\ulcorner-algebra into a commutative radical nil Banach algebra.

• 충청수학회지
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• 제11권1호
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• pp.151-161
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• 1998

The odd spherical non-commutative tori $\mathbb{S}_{\omega}$ were defined in [2]. Assume that no non-trivial matrix algebra can be factored out of $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_{km}(\mathbb{C})$. It is shown that the tensor product of $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has the trivial bundle structure if and, only if km and 2d - 1 are relatively prime, and that the tensor product of $\mathbb{S}_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when km > 1.

• Journal of applied mathematics & informatics
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• 제15권1_2호
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• pp.343-361
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• 2004

We prove the generalized Hyers-Ulam-Rassias stability of cyclic functional equations in Banach modules over a unital $C^{*}$-algebra. It is applied to show the stability of algebra homomorphisms between Banach algebras associated with cyclic functional equations in Banach algebras.

• 대한수학회지
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• 제41권3호
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• pp.461-477
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• 2004

It is shown that every almost linear mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A of real rank zero to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of *-homomorphisms between unital $C^{*}$ -algebras, and of C-linear *-derivations on unital $C^{*}$ -algebras./ -algebras.