• Title/Summary/Keyword: algebra

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POSITIVE LINEAR OPERATORS IN C*-ALGEBRAS

  • Park, Choon-Kil;An, Jong-Su
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.5
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    • pp.1031-1040
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    • 2009
  • It is shown that every almost positive linear mapping h : $\mathcal{A}\rightarrow\mathcal{B}$ of a Banach *-algebra $\mathcal{A}$ to a Banach *-algebra $\mathcal{B}$ is a positive linear operator when h(rx) = rh(x) (r > 1) holds for all $x\in\mathcal{A}$, and that every almost linear mapping h : $\mathcal{A}\rightarrow\mathcal{B}$ of a unital C*-algebra $\mathcal{A}$ to a unital C*-algebra $\mathcal{B}$ is a positive linear operator when h($2^nu*y$) = h($2^nu$)*h(y) holds for all unitaries $u\in \mathcal{A}$, all $y \in \mathcal{A}$, and all n = 0, 1, 2, ..., by using the Hyers-Ulam-Rassias stability of functional equations. Under a more weak condition than the condition as given above, we prove that every almost linear mapping h : $\mathcal{A}\rightarrow\mathcal{B}$ of a unital C*-algebra $\mathcal{A}$ A to a unital C*-algebra $\mathcal{B}$ is a positive linear operator. It is applied to investigate states, center states and center-valued traces.

A Didactical Discussion on the Use of Mathematical Manipulatives (교구이용에 대한 교수학적 논의 -대수모델의 활용사례를 통한 교구의 효과 분석을 중심으로-)

  • 김남희
    • School Mathematics
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    • v.2 no.1
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    • pp.29-51
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    • 2000
  • In this study, we tried to suggest an example of the analysis on the use of mathematical manipulatives. Taking algebra tiles as an example of mathematical manipulatives, we analysed several effects resulted from the use of algebra tiles. The algebra tiles make it possible to do activities that are needed to introduce and explain the distributive law and factoring. The algebra tiles have a several advantages; First of all, This model is simple. Even though they cannot make algebra easy, this model can play an important role in the transition to a new algebra course. This model provides access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense. This model provides a geometric interpretation of symbol manipulation, thereby enriching students' understanding, This model supports cooperative learning, and help improve discourse in the algebra class by giving students objects to think with and talk about. On the other hand, The disadvantages of this model are as follows; the model reinforces the misconception that -x is negative, and x is positive; the area model of multiplication is not geometrically sound when minus is involved; only the simplest expressions involving minus can be represented; It is ineffective when be used the learning of already known concept. Mathematics teachers must have a correct understanding about these advantages and disadvantages of manipulatives. Therefore, they have to plan classroom work that be maximized the positive effect of manipulatives and minimized the negative effect of manipulatives.

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BANACH FUNCTION ALGEBRAS OF n-TIMES CONTINUOUSLY DIFFERENTIABLE FUNCTIONS ON Rd VANISHING AT INFINITY AND THEIR BSE-EXTENSIONS

  • Inoue, Jyunji;Takahasi, Sin-Ei
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1333-1354
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    • 2019
  • In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

SCALAR EXTENSION OF SCHUR ALGEBRAS

  • Choi, Eun-Mi
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.453-467
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    • 2005
  • Let K be an algebraic number field. If k is the maximal cyclotomic subextension in K then the Schur K-group S(K) is obtained from the Schur k-group S(k) by scalar extension. In the paper we study projective Schur group PS(K) which is a generalization of Schur group, and prove that a projective Schur K-algebra is obtained by scalar extension of a projective Schur k-algebra where k is the maximal radical extension in K with mild condition.

NOTES ON ${\overline{WN_{n,0,0_{[2]}}}$ II

  • CHOI, SEUL HEE
    • Honam Mathematical Journal
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    • v.27 no.4
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    • pp.583-593
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    • 2005
  • The Weyl-type non-associative algebra ${\overline{WN_{g_n,m,s_r}}$ and its subalgebra ${\overline{WN_{n,m,s_r}}$ are defined and studied in the papers [2], [3], [9], [11], [12]. We find the derivation group $Der_{non}({\overline{WN_{1,0,0_{[2]}}})$ the non-associative simple algebra ${\overline{WN_{1,0,0_{[2]}}}$.

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ON SUBREGULAR POINTS FOR SOME CASES OF LIE ALGEBRA

  • KIM, Y.K.;SO, K.H.;SEO, G.S.;PARK, D.Y.;CHOI, S.H.
    • Honam Mathematical Journal
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    • v.19 no.1
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    • pp.21-27
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    • 1997
  • We shall define three kinds of points for algebraic varieties associated to the center 3 of U(L) which is the universal enveloping algebra of a finite-dimensional modular Lie algebra over an algebraically closed field F of prime characteristic p. We announce here that $sp_4$(F) with p = 2 has a subregular point.

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THE STABILITY OF A DERIVATION ON A BANACH ALGEBRA

  • LEE, EUN HWI;CHANG, ICK-SOON;JUNG, YONG-SOO
    • Honam Mathematical Journal
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    • v.28 no.1
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    • pp.113-124
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    • 2006
  • In this article, we show that for an approximate derivation on a Banach *-algebra, there exist a unique derivation near the an approximate derivation and for an approximate derivation on a $C^*$-algebra, there exist a unique derivation near the approximate derivation.

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GENERALIZED BIPRODUCT HOPF ALGEBRAS

  • Park, Junseok
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.3
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    • pp.301-320
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    • 2008
  • The smash product algebra has been generalized to general smash product algebra in [3] and we can generalize the smash coproduct coalgebra to obtain the general smash coproduct coalgebra. It is natural to replace the smash product and smash coproduct by the generalized smash product and generalized smash coproduct and consider the condition under which the generalized smash product algebra structure and the generalized smash coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary sufficient conditions for the problem. This generalizes the corresponding results in [7] and [4].

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REGULAR GLOSED BOOLEAN ALGBRA IN THE SPACE WITH EXTENSION TOPOLOGY

  • Cao, Shangmin
    • The Pure and Applied Mathematics
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    • v.7 no.2
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    • pp.71-78
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    • 2000
  • Any Hausdoroff space on a set which has at least two points a regular closed Boolean algebra different from the indiscrete regular closed Boolean algebra as indiscrete space. The Sierpinski space and the space with finite complement topology on any infinite set etc. do the same. However, there is $T_{0}$ space which does the same with Hausdorpff space as above. The regular closed Boolean algebra in a topological space is isomorphic to that algebra in the space with its open extension topology.

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