• Korean Journal of Mathematics
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• v.27 no.3
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• pp.735-741
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• 2019

Let ${\mathcal{H}}$ be a complex Hilbert space and ${\mathcal{B}}({\mathcal{H}})$ the algebra of all bounded linear operators on ${\mathcal{H}}$. In this paper, we prove that if ${\varphi}:{\mathcal{B}}({\mathcal{H}}){\rightarrow}{\mathcal{B}}({\mathcal{H}})$ is a unital surjective bounded linear map, which preserves m- isometries m = 1, 2 in both directions, then there are unitary operators $U,V{\in}{\mathcal{B}}({\mathcal{H}})$ such that ${\varphi}(T)=UTV$ or ${\varphi}(T)=UT^{tr}V$ for all $T{\in}{\mathcal{B}}({\mathcal{H}})$, where $T^{tr}$ is the transpose of T with respect to an arbitrary but fixed orthonormal basis of ${\mathcal{H}}$.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.375-381
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• 2019

In this paper, we introduce an operation denoted by [$Br_e$], a bracket operation, which maps an arbitrary groupoid ($X,{\ast}$) on a set X to another groupoid $(X,{\bullet})=[Br_e](X,{\ast})$ which on groups corresponds to sending a pair of elements (x, y) of X to its commutator $xyx^{-1}y^{-1}$. When applied to classes such as d-algebras, BCK-algebras, a variety of results is obtained indicating that this construction is more generally useful than merely for groups where it is of fundamental importance.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.1181-1197
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• 2022

We give a complete classification of simply connected and solvable real Lie groups whose nontrivial coadjoint orbits are of codimension 1. This classification of the Lie groups is one to one corresponding to the classification of their Lie algebras. Such a Lie group belongs to a class, called the class of MD-groups. The Lie algebra of an MD-group is called an MD-algebra. Some interest properties of MD-algebras will be investigated as well.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.957-967
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• 2022

Recently, Brešar's Jordan {g, h}-derivations have been investigated on triangular algebras. As a first aim of this paper, we extend this study to an interesting general context. Namely, we introduce the notion of Jordan 𝒢_n-derivations, with n ≥ 2, which is a natural generalization of Jordan {g, h}-derivations. Then, we study this notion on path algebras. We prove that, when n > 2, every Jordan 𝒢_n-derivation on a path algebra is a {g, h}-derivation. However, when n = 2, we give an example showing that this implication does not hold true in general. So, we characterize when it holds. As a second aim, we give a positive answer to a variant of Lvov-Kaplansky conjecture on path algebras. Namely, we show that the set of values of a multi-linear polynomial on a path algebra KE is either {0}, KE or the space spanned by paths of a length greater than or equal to 1.

• 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.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.331-343
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• 2024

Let ℜ be a *-algebra with unity I and a nontrivial projection P₁. In this paper, we show that under certain restrictions if a map ψ : ℜ → ℜ satisfies $$\Psi(S_1{\diamond}S_2{\cdot}{\cdot}{\cdot}{\diamond}S_{n-1}{\bullet}S_n)=\sum_{k=1}^nS_1{\diamond}S_2{\diamond}{\cdot}{\cdot}{\cdot}{\diamond}S_{k-1}{\diamond}{\Psi}(S_k){\diamond}S_{k+1}{\diamond}{\cdot}{\cdot}{\cdot}{\diamond}S_{n-1}{\bullet}S_n$$ for all S_n-2, S_n-1, S_n ∈ ℜ and S_i = I for all i ∈ {1, 2, . . . , n - 3}, where n ≥ 3, then ψ is an additive *-derivation.

• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.71-80
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• 2001

The purpose of this paper is to give the classification of homogeneous integral table algebras of degree 5 containing a faithful real element of which 2. In fact, these algebras are classified to exact isomorphism, that is the sets of structure constants which arise from the given basis are completely determined. This is work towards classifying homogeneous integral table algebras of degree 5. AMS Mathematics Subject Classification : 20C05, 20C99.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.751-755
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• 2008

Formanek([2]) proved that $M_n(K)$, the matrix algebra has a nontrivial central polynomial when char K = 0. Also Razmyslov([3]) showed the same result using the essential weak identity. In this article we explicitly compute Formanek's central polynomial for $M_2(\mathbb{C})$ and $M_3(\mathbb{C})$ and classify the coefficients of the central polynomial.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1605-1614
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• 2014

In continuation to the investigation initiated by Ferraz, Goodaire and Milies in [4], we provide an explicit description for the Wedderburn decomposition of finite semisimple group algebras of the class of finite groups G, such that $$G/Z(G){\simeq_-}C_2{\times}C_2$$, where Z(G) denotes the center of G.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.621-629
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• 2008

Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and $\delta$ a generalized derivations of R. If [[$\delta(x)$, x], $\delta(x)$] = 0 for all $x\;{\in}\;R$ then either R is commutative or $\delta(x)\;=\;ax$ for all $x\;{\in}\;R$ and some $a\;{\in}\;C$. We also obtain some related result in case R is a Banach algebra and $\delta$ is either continuous or spectrally bounded.