• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.189-198
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• 2004

In this study we consider quantum dynamical semi-group with a normal faithful invariant state. A quantum dynamical semigroup $\alpha\;=\;\{{\alpha}_t\}_{t{\geq}0}$ is a class of linear normal identity-preserving mappings on a von Neumann algebra M with semigroup property and some positivity condition. We investigate the asymptotic behaviors of the semigroup such as ergodicity or mixing properties in terms of their eigenvalues under the assumption that the semigroup satisfies positivity. This extends the result of [13] which is obtained under the assumption that the semi group satisfy 2-positivity.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.637-657
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• 2021

In this paper, we decompose (under unitary similarity) the Kronecker sum A ⊞ A (= A ⊗ I + I ⊗ A) into a direct sum of irreducible matrices, when A is a 3×3 matrix. As a consequence we identify 𝒦(A⊞A) as the direct sum of several full matrix algebras as predicted by Artin-Wedderburn theorem, where 𝒦(T) is the unital algebra generated by Tand T^*.

• Nonlinear Functional Analysis and Applications
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• v.26 no.4
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• pp.649-662
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• 2021

Recently the author studied a one-parameter family of divergences and considered the related median minimization problem of finite points over these divergences in general symmetric cones. In this article, to utilize the results practically, we deal with a special symmetric cone called second order cone, which is important in optimization fields. To be more specific, concrete computations of divergences with its gradients and the unique minimizer of the median minimization problem of two points are provided skillfully.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1137-1150
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• 2022

Consider the quantum algebra U_q(sl₂) over field 𝓕 (char(𝓕) = 0) with equitable generators x^±1, y and z, where q is fixed nonzero, not root of unity scalar in 𝓕. Let V denote a finite dimensional irreducible module for this algebra. Let Λ ∈ End(V), and let {A₁, A₂, A₃} = {x, y, z}. First we show that if Λ, A₁ is a Leonard pair, then this Leonard pair have four types, and we show that for each type there exists a Leonard pair Λ, A₁ in which Λ is a linear combination of 1, A₂, A₃, A₂A₃. Moreover, we use Λ to construct 𝚼 ∈ U_q(sl₂) such that 𝚼, A^-1₁ is a Leonard pair, and show that 𝚼 = I + A₁Φ + A₁ΨA₁ where Φ and Ψ are linear combination of 1, A₂, A₃.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.635-643
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• 2002

Our main goal is to show that if there exist Jordan derivations D and G on a noncommutative (n + 1)!-torsion free prime ring R such that $$D(x)x^n-x^nG(x)\in\ C(R)$$ for all $x\in\ R$, then we have D=0 and G=0. We also prove that if there exists a derivation D on a noncommutative 2-torsion free prime ring R such that the mapping $\chi$longrightarrow[aD($\chi$), $\chi$] is commuting on R, then we have either a = 0 or D = 0.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.401-414
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• 2019

Let ${\mathcal{F}}$ denote an algebraically closed field with characteristic not two. Fix an integer $d{\geq}3$, let $Mat_{d+1}({\mathcal{F}})$ denote the ${\mathcal{F}}$-algebra of $(d+1){\times}(d+1)$ matrices with entries in ${\mathcal{F}}$. An ordered pair of matrices A, $A^*$ in $Mat_{d+1}({\mathcal{F}})$ is said to be LB-TD form whenever A is lower bidiagonal with subdiagonal entries all 1 and $A^*$ is irreducible tridiagonal. Let A, $A^*$ be a Leonard pair in $Mat_{d+1}({\mathcal{F}})$ with fundamental parameter ${\beta}=2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.

• Bulletin of the Korean Mathematical Society
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• v.23 no.2
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• pp.167-170
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• 1986

Let M be a von Neumann algebra and .alpha., .betha. be *-automorphisms of M satisfying the operator equation .alpha.+.alpha.$^{-1}$ =.betha.+.betha.$^{-1}$ This operator equation has been extensively studied and many important decomposition theorems have been obtained by several authors (for instance see [4], [5], [2], [1]). Originally, this operator equation arose in the paper of Van Daele on the new approach of the Tomita-Takesaki theory in the case of modular operators ([7]). In the case of one-parameter automorphism groups, this equation has produced a bounded and completely positive map which can play a role similar to the infinitesimal generator (for details see [6] and [1]). A recent and one of the most important applications of this equation has been in developing an anglogue of the Tomita-Takesaki theory for Jordan algebras by Haagerup [3]. One general result of this theory is the following.