• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1019-1028
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• 2023

Let 𝔄 and 𝔅 be unital prime *-algebras such that 𝔄 contains a nontrivial projection. In the present paper, we show that if a bijective map Θ : 𝔄 → 𝔅 satisfies Θ(_*[X ⋄ Y, Z]) = _*[Θ(X) ⋄ Θ(Y), Θ(Z)] for all X, Y, Z ∈ 𝔄, then Θ or -Θ is a *-ring isomorphism. As an application, we shall characterize such maps in factor von Neumann algebras.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.653-659
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• 1995

We prove that two essentially normal elements of a type $II_{\infty}$ factor von Neumann algebra are unitarily equivalent up to the compact ideal if and only if they have the identical essential spectrum and the same index data. Also we calculate the spectrum and essential spectrum of a non-unitary isometry of von Neumann algebra.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.581-591
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• 2015

Let $\mathcal{A}$ be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map ${\delta}:\mathcal{A}{\rightarrow}\mathcal{A}$ satisfies $${\delta}([[a,b],c])=[[{\delta}(a),b],c]+[[a,{\delta}(b),c]+[[a,b],{\delta}(c)]$$ for any $a,b,c{\in}\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of $\mathcal{A}$), then there exist an operator $T{\in}\mathcal{A}$ and a linear map $f:\mathcal{A}{\rightarrow}\mathbb{C}I$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that ${\delta}(a)=aT-Ta+f(a)$ for any $a{\in}\mathcal{A}$.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.459-465
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• 2018

Recently, Li et al proved that ${\Phi}$ which satisfies the following condition on factor von Neumann algebras $${\Phi}([[A,B]_*,C]_*)=[[{\Phi}(A),B]_*,C]_*+[[A,{\Phi}(B)]_*,C]_*+[[A,B]_*,{\Phi}(C)]_*$$ where $[A,B]_*=AB-BA^*$ for all $A,B{\in}{\mathcal{A}}$, is additive ${\ast}-derivation$. In this short note we show the additivity of ${\Phi}$ which satisfies the above condition on prime ${\ast}-algebras$.

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.29-36
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• 1998

We show that if M is a $II_1$-factor and a countable discrete group G acts outerly on M then Jones' index $[M:M^G]$ of a pair of $II_1^-factors is equal to the order $\mid$G$\mid$ of G. It is also shown that for a subgroup H of G Jones' index $[M^H:M^G]$ is equal to the group index [G:H] under certain conditions.

• Bulletin of the Korean Mathematical Society
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• v.23 no.2
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• pp.117-121
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• 1986

Connes [2] showed that if M is an injective .sigma.-finite factor of type II $I_{1}$ and $B_{\phi}$=C1 for some normal faithful state .phi. on M then M is isomorphic to the Araki-wood factor. In [7], Haagerup has succeeded to show that if M is an injective factor of type II $I_{1}$ with separable predual, then $B_{\phi}$=C1 for every normal faithful state on M. Since injective factors of type II $I_{\lambda}$, 0.leq..lambda.<1, were classified [9], this together classifies all injective factors of type III with separable predual. It is not known for non-injective case. In this paper we consider some conditions under which the bicentralizers be trivial.ivial.