• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.293-299
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• 2005

Let ${\cal{L}}$ be a commutative subspace lattice on a Hilbert space ${\cal{H}}$ and X and Y be operators on ${\cal{H}}$. Let $${\cal{M}}_X=\{{\sum}{\limits_{i=1}^n}E_{i}Xf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}$$ and $${\cal{M}}_Y=\{{\sum}{\limits_{i=1}^n}E_{i}Yf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}.$$ Then the following are equivalent. (i) There is an operator A in $Alg{\cal{L}}$ such that AX = Y, Ag = 0 for all g in ${\overline{{\cal{M}}_X}}^{\bot},A^*A=AA^*$ and every E in ${\cal{L}}$ reduces A. (ii) ${\sup}\;\{K(E, f)\;:\;n\;{\in}\;{\mathbb{N}},f_i\;{\in}\;{\cal{H}}\;and\;E_i\;{\in}\;{\cal{L}}\}\;<\;\infty,\;{\overline{{\cal{M}}_Y}}\;{\subset}\;{\overline{{\cal{M}}_X}}$and there is an operator T acting on ${\cal{H}}$ such that ${\langle}EX\;f,Tg{\rangle}={\langle}EY\;f,Xg{\rangle}$ and ${\langle}ET\;f,Tg{\rangle}={\langle}EY\;f,Yg{\rangle}$ for all f, g in ${\cal{H}}$ and E in ${\cal{L}}$, where $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.167-170
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• 1988

Versions of Tannaka duality in operator algebraic context have been obtained in [6], [8] etc. Suppose .sigma.is an automorphism of a von Neumann algebra M, on which there is an action .alpha. of a compact group G such that .sigma. vertical bar $M^{\alpha}$=id, where $M^{\tau}$is the fixed point algebra under the action .alpha.. Then it is shown that if there is an action .tau. of a group H which commutes with .alpha., and which is ergodic in the sense that the fixed point algebra $M^{\tau}$ is trivial, then there exists g.mem.G such that .sigma.=.alpha.(g). Recently Evans and Kishimoto ([4]) showed the versions of Tannaka duality in $C^{*}$-settings under some conditions.s.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.91-95
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• 1995

Let H be an arbitrary complex Hilbert space and let (equation omitted)(H) be the *-algebra of all bounded linear operators on H. An operator T in (equation omitted)(H) is called normal if T$\^$＊/T = TT$\^$＊/, hyponormal if T$\^$＊/T $\geq$ TT$\^$＊/, and quasi-hyponormal if T$\^$＊/(T$\^$＊/T － TT$\^$＊/)A $\geq$ 0, or equivalently ∥T$\^$＊/T$\chi$∥ $\leq$ ∥TT$\chi$∥ for all $\chi$ in H.(omitted)

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.597-609
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• 2008

In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space Hand G, a group and let ${\alpha}$ : G${\rightarrow}$ AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product $\mathbb{M}=M{\times}{\alpha}$ G of M and G with respect to ${\alpha}$ is a von Neumann algebra acting on $H{\bigotimes}{\iota}^2(G)$, generated by M and $(u_g)_g{\in}G$, where $u_g$ is the unitary representation of g on ${\iota}^2(G)$. We show that $M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)$. We compute moments and cumulants of operators in $\mathbb{M}$. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if $F_N$ is the free group with N-generators, then the crossed product algebra $L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n$ satisfies that $$L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})$$, whenerver $n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.541-548
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• 2006

We find a graded Artinian level O-sequence of the form $H\;:\;h_0\;h_1\;\cdots\;h_{d-1}\;h_d\cdots$ $^{(d+1-1_)-st}h_d$ < $h_{d+s}$ not having the Weak-Lefschetz property. We also introduce several algorithms for construction of some examples of non-unimodal level O-sequences using a computer program called CoCoA.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.363-372
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• 2023

In this paper, we consider two functional equations: (1) h(𝓕(x, y, z) + 2x + y + z) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) + 2x + y) + h(xy) + yh(x + z) + 2h(z), (2) h(𝓕(x, y, z) - y + z + 2e) + 2h(x + y) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) - y + 2e) + 2h(x + y + z) + h(xy) + yh(x + z), without any regularity assumption for all x, y, z in a unital algebra A, where 𝓕 : A³ → A is defined by 𝓕(x, y, z) := h(x + y + z) - h(x + y) - h(z) for all x, y, z ∈ A. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.

• East Asian mathematical journal
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• v.22 no.2
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• pp.249-257
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• 2006

Let n be a 2-step nilpotent Lie algebra which has an inner product <,> and has an orthogonal decomposition $n=\delta{\oplus}\varsigma$ for its center $\delta$ and the orthogonal complement $\varsigma\;of\;\delta$. Then Each element Z of $\delta$ defines a skew symmetric linear map $J_Z:\varsigma{\rightarrow}\varsigma$ given by $=$ for all $X,\;Y{\in}\varsigma$. Let $\gamma$ be a unit speed geodesic in a 2-step nilpotent Lie group H(2, 1) with its Lie algebra n(2, 1) and let its initial velocity ${\gamma}$(0) be given by ${\gamma}(0)=Z_0+X_0{\in}\delta{\oplus}\varsigma=n(2,\;1)$ with its center component $Z_0$ nonzero. Then we showed that $\gamma(0)$ is conjugate to $\gamma(\frac{2n{\pi}}{\theta})$, where n is a nonzero intger and $-{\theta}^2$ is a nonzero eigenvalue of $J^2_{Z_0}$, along $\gamma$ if and only if either $X_0$ is an eigenvector of $J^2_{Z_0}$ or $adX_0:\varsigma{\rightarrow}\delta$ is not surjective.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.109-113
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• 1995

For a Hilbert space H, let L(H) denote the algebra of all bounded operators on H. For an $n \in N$, it is well known that any element $T \in L(\oplus^n H)$ is expressed as an $n \times n$ matrix each of whose entries lies in L(H) so that T is written as $$ (1) T = (T_{ij}), i, j = 1, 2, ..., n, T_{ij} \in L(H), $$ where $\oplus^n H$ is the direct sum Hilbert space of n copies of H.

• Journal of the Chungcheong Mathematical Society
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• v.7 no.1
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• pp.117-122
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• 1994

In this paper we show that if {$H_n$} is a continuous strong higher derivation of order n on an ultraprime Banach algebra with a constant c, then $c||H_1||^2{\leq}4||H_2||$ and for each $1{\leq}l$ < n $$c^2||H_1||\;||H_{n-l}{\leq}6||H_n||+\frac{3}{2}\sum_{\array{i+j+k=n\\i,j,k{\geq}1}}||H_i||\;||H_j||\;||H_k||+\frac{3}{2}\sum_{\array{i+k=n\\i{\neq}l,\;n-1}}||H_i||\;||H_k|| $$ and for a strong higher derivation {$H_n$} of order n on a prime ring A we also show that if [$H_n$(x),x]=0 for all $x{\in}A$ and for every $n{\geq}1$, then A is commutative or $H_n=0$ for every $n{\geq}1$.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.127-135
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n＋l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)＋as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.