• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.193-199
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• 1993

A hereditary $C^{*}$-subalgebra B of a $C^{*}$-algebra A is said to be full if B is not contained in any proper closed two-sided ideal in A, so each hereditary $C^{*}$-subalgebra of a simple $C^{*}$-algebra is always full. It is well known that every $C^{*}$-algebra is strong Morita equivalent to its full hereditary $C^{*}$-subalgebra, but the strong Morita equivalence of a $C^{*}$-algebra A and its hereditary $C^{*}$-subalgebra B does not imply the fullness of B, ingeneral. We present the following lemma for our computational convenience in the course of the proof of the main theorem. Note that $L_{B}$, $L_{B}$$^{*}$ and $L_{B}$ $L_{B}$$^{*}$ are all .alpha.-invariant whenever B is .alpha.-invariant under the action .alpha. of G.a. of G.a. of G.a. of G.f G.

• The Pure and Applied Mathematics
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• v.28 no.3
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• pp.217-234
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• 2021

In this paper, first we introduce the new class of HV-BE-algebra as a generalization of a (hyper) BE-algebra and prove some basic results and present several examples. Then, we construct the HV-BE-algebra associated to a BE-algebra (namely BL-BE-algebra) based on "Begins lemma" and investigate it.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.359-366
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• 1996

A linear map T from a Banach algebra A into a Banach algebra B is almost multiplicative if $\left\$\mid$ T(fg) - T(f)T(g) \right\$\mid$ \leq \in\left\$\mid$ f \right\$\mid$\left\$\mid$ g \right\$\mid$(f,g \in A)$ for some small positive $\in$. B.E.Johnson [4,5] studied whether this implies that T is near a multiplicative map in the norm of operators from A into B. K. Jarosz [2,3] raised the conjecture : If T is an almost multiplicative functional on uniform algebra A, there is a linear and multiplicative functional F on A such that $\left\$\mid$ T - F \right\$\mid$ \leq \in', where \in' \to 0$ as $\in \to 0$. B. E. Johnson [4] gave an example of non-uniform commutative Banach algebra which does not have the property described in the above conjecture. He proved also that C(K) algebras and the disc algebra A(D) have this property [5]. We extend this property to a derivation on a Banach algebra.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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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.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1129-1135
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• 2022

The purpose of this research is to find an extension of the renowned Chernoff theorem on standard operator algebra. Infact, we prove the following result: Let H be a real (or complex) Banach space and 𝓛(H) be the algebra of bounded linear operators on H. Let 𝓐(H) ⊂ 𝓛(H) be a standard operator algebra. Suppose that D : 𝓐(H) → 𝓛(H) is a linear mapping satisfying the relation D(AⁿBⁿ) = D(Aⁿ)Bⁿ + AⁿD(Bⁿ) for all A, B ∈ 𝓐(H). Then D is a linear derivation on 𝓐(H). In particular, D is continuous. We also present the limitations on such identity by an example.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.93-113
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• 2011

Let $a{\oplus}b$ = max(a, b), $a{\otimes}b$=a+b, a, $b\in\mathbb{R}_{\varepsilon}\;:=\cup\{-\infty\}$. In max-plus algebra we work on the linear algebra structure for the pair of operations (${\oplus},{\otimes}$) extended to matrices and vectors over $\mathbb{R}_{\varepsilon}$. In this paper our main aim is to reproduce the work of R. A. Cuninghame-Green [3] on the linear systems over a max-plus semi-field $\mathbb{R}_{\varepsilon}$.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.285-291
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• 2012

Let $M_n$ be the algebra of all complex $n{\times}n$ matrices and ${\phi}:M_n{\rightarrow}M_n$ a surjective map (not necessarily additive or multiplicative) satisfying one of the following equations: $${\det}({\phi}(A){\phi}(B)+{\phi}(X))={\det}(AB+X),\;A,B,X{\in}M_n,\\{\sigma}({\phi}(A){\phi}(B)+{\phi}(X))={\sigma}(AB+X),\;A,B,X{\in}M_n$$. Then it is an automorphism, where ${\sigma}(A)$ is the spectrum of $A{\in}M_n$. We also show that if $\mathfrak{A}$ be a standard operator algebra, $\mathfrak{B}$ is a unital Banach algebra with trivial center and if ${\phi}:\mathfrak{A}{\rightarrow}\mathfrak{B}$ is a multiplicative surjection preserving spectrum, then ${\phi}$ is an algebra isomorphism.

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.241-250
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• 2003

Let (B, m$_{B}$, k) be a maximal commutative $textsc{k}$-subalgebra of M$_{m}$(k). Then, for some element z $\in$ Soc(B), a k-algebra R = B［X,Y］/I, where I ＝ (m$_{B}$X, m$_{B}$Y, X$^2$- z,Y$^2$- z, XY) will create an interesting maximal commutative $textsc{k}$-subalgebra of a matrix algebra which is neither a $C_1$-construction nor a $C_2$-construction. This construction will also be useful to embed a maximal commutative $textsc{k}$-subalgebra of matrix algebra to a maximal commutative $textsc{k}$-subalgebra of a larger size matrix algebra.gebra.a.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.253-259
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• 2011

Let A and B be Banach algebras with unit. Here we prove that an approximate algebra homomorphism f : A ${\rightarrow}$ B, in the sense of Rassias, is an algebra homomorphism.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.613-629
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• 2016

In this paper, we introduced the notion of multiplier of Boolean algebras and discuss related properties between multipliers and special mappings, like dual closures, homomorphisms on B. We introduce the notions of xed set $Fix_f(X)$ and normal ideal and obtain interconnection between multipliers and $Fix_f(B)$. Also, we introduce the special multiplier ${\alpha}_p$a nd study some properties. Finally, we show that if B is a Boolean algebra, then the set of all multipliers of B is also a Boolean algebra.