• 대한수학회보
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• 제31권2호
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• pp.173-180
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• 1994

Let H be a separable, infinite dimensional, compled Hilbert space and let L(H) be the algebra of all bounded linear operators on H. A dual algebra is a subalgebra of L(H) that contains the identity operator $I_{H}$ and is closed in the ultraweak topology on L(H). Note that the ultraweak operator topology coincides with the wea $k^{*}$ topology on L(H)(see [3]). Bercovici-Foias-Pearcy [3] studied the problem of solving systems of simultaneous equations in the predual of a dual algebra. The theory of dual algebras has been applied to the topics of invariant subspaces, dilation theory and reflexibity (see [1],[2],[3],[5],[6]), and is deeply related with properties ( $A_{m,n}$). Jung-Lee-Lee [7] introduced n-separating sets for subalgebras and proved the relationship between n-separating sets and properties ( $A_{m,n}$). In this paper we will study the relationship between direct sum and properties ( $A_{m,n}$). In particular, using some results of [7] we obtain relationship between n-separating sets and direct sum of von Neumann algebras.ras.s.ras.

• 대한수학회논문집
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• 제18권3호
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• pp.439-447
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• 2003

Let (equation omitted) be a symmetrizable Kac-Moody algebra with the indecomposable generalized Cartan matrix A and W be its Weyl group. Let $\theta$ be the highest root of the corresponding finite dimensional simple Lie algebra ${\gg}$ of g. For the type ${A_N}^{(r)}$, we give an element $\omega_{o}\;\in\;W$ such that ${{\omega}_o}^{-1}({\{\Delta\Delta}_{+}})\;=\;{\{\Delta\Delta}_{-}}$. And then we prove that the degree of nilpotency of the subalgebra (equation omitted) is greater than or equal to $ht{\theta}+1$.

• 대한수학회지
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• 제31권4호
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• pp.659-667
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• 1994

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ be the algebra of all bounded linear operators on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $I_H$ and is closed in the ultraweak operator topology on $L(H)$. Note that the ultraweak operator topology coincides with the weak topology on $L(H) (cf. [6]). Several functional analysists have studied the problem of solving systems of simultaneous equations in the predual of a dual algebra (cf. [3]). This theory is applied to the study of invariant subspaces and dilation theory, which are deeply related to the classes $A_{m,n}$ (that will be defined below) (cf. [3]). An abstract geometric criterion for dual algebras with property $(A_{\aleph_0}, {\aleph_0})$ was first given in [1].

• 대한수학회보
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• 제44권1호
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• pp.131-150
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• 2007

In this paper, we will define direct producted $W^*-porobability$ spaces over their diagonal subalgebras and observe the amalgamated free-ness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let ($A_{j},\;{\varphi}_{j}$) be a tracial $W^*-porobability$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $W^*-porobability$ space (A, E) over its N-th diagonal subalgebra $D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}$, where $A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}$. In Chapter 1, we show that $D_{N}-valued$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $D_{N}-freeness$ is characterized by the direct sum of scalar-valued freeness. As application, the $D_{N}-semicircularityrity$ and the $D_{N}-valued$ infinitely divisibility are characterized by the direct sum of semicircularity and the direct sum of infinitely divisibility, respectively. In Chapter 2, we will define the $D_{N}-valued$ stochastic integral of $D_{N}-valued$ simple adapted biprocesses with respect to a fixed $D_{N}-valued$ infinitely divisible element which is a $D_{N}-free$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $D_{N}-valued$ case.

• Journal of applied mathematics & informatics
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• 제10권1_2호
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• pp.51-62
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• 2002

The present paper gives a new construction of a quotient BCI(BCK)-algebra X/${\mu}$ by a fuzzy ideal ${\mu}$ in X and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if ${\mu}$ is a fuzzy ideal (closed fuzzy ideal) of X, then X/${\mu}$ is a commutative (resp. positive implicative, implicative) BCK(BCI)-algebra if and only if It is a fuzzy commutative (resp. positive implicative, implicative) ideal of X Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra of X We show that if the period of every element in a BCI-algebra X is finite, then any fuzzy ideal of X is closed. Especiatly, in a well (resp. finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed.