• Title, Summary, Keyword: von Neumann algebra

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ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA

  • Cho, Il-Woo
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.601-631
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    • 2010
  • In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar-valued $W^*$-probability on a given graph von Neumann algebra. We show that a diagonal graph $W^*$-probability space (as a scalar-valued $W^*$-probability space) and a graph W¤-probability space (as an amalgamated $W^*$-probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

TYPE $I_{\infty}$ OF A VON NEUMANN ALGEBRA ALG$\mathcal{L}$

  • Kim, Jong-Geon
    • East Asian mathematical journal
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    • v.15 no.2
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    • pp.313-324
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    • 1999
  • What we will be concerned with is, first, the question of the condition about $\mathcal{L}$ that gives Alg$\mathcal{L}$ a von Neumann algebra, that is, the question of the condition about $\mathcal{L}$ that will give Alg$\mathcal{L}$ a self-adjoint algebra. Secondly, if Alg$\mathcal{L}$ is a von Neumann algebra, we want to find out what type it is.

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Essentially normal elements of von neumann algebras

  • Cho, Sung-Je
    • 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.

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CONSTRUCTION OF UNBOUNDED DIRICHLET FOR ON STANDARD FORMS OF VON NEUMANN ALGEBRAS

  • Bahn, Chang-Soo;Ko, Chul-Ki
    • Journal of the Korean Mathematical Society
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    • v.39 no.6
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    • pp.931-951
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    • 2002
  • We extend the construction of Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebra given in [13] to the case of unbounded operators satiated with the von Neumann algebra. We then apply our result to give Dirichlet forms associated to the momentum and position operators on quantum mechanical systems.

PROPERTY T FOR FINITE VON NEUMANN ALGEBRAS

  • Boo, Deok-Hoon;Park, Chun-Gil
    • Journal of the Chungcheong Mathematical Society
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    • v.10 no.1
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    • pp.117-126
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    • 1997
  • We find more simple forms of property T for von Neumann algebras which are finite direct sum of $II_1$ factors.

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CHARACTERIZATIONS OF CENTRALIZERS AND DERIVATIONS ON SOME ALGEBRAS

  • He, Jun;Li, Jiankui;Qian, Wenhua
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.685-696
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    • 2017
  • A linear mapping ${\phi}$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G{\in}{\mathcal{A}}$ if ${\phi}(AB)={\phi}(A)B= A{\phi}(B)$ for each A and B in $\mathcal{A}$ with AB = G, and ${\phi}$ is called a derivable mapping at $G{\in}{\mathcal{A}}$ if ${\phi}(AB)={\phi}(A)B+A{\phi}(B)$ for each A and B in $\mathcal{A}$ with AB = G. A point G in A is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at G is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.

SELF-ADJOINT CYCLICALLY COMPACT OPERATORS AND ITS APPLICATION

  • Kudaybergenov, Karimbergen;Mukhamedov, Farrukh
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.679-686
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    • 2017
  • The present paper is devoted to self-adjoint cyclically compact operators on Hilbert-Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators is given. We use more simple and constructive method, which allowed to apply this result to compact operators relative to von Neumann algebras. Namely, a general form of compact operators relative to a type I von Neumann algebra is given.

NONLINEAR ξ-LIE-⁎-DERIVATIONS ON VON NEUMANN ALGEBRAS

  • Yang, Aili
    • Korean Journal of Mathematics
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    • v.27 no.4
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    • pp.969-976
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    • 2019
  • Let ℬ(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ and �� ⊆ ℬ(ℋ) be a von Neumann algebra without central abelian projections. Let ξ be a non-zero scalar. In this paper, it is proved that a mapping φ : �� → ℬ(ℋ) satisfies φ([A, B]ξ⁎)= [φ(A), B]ξ⁎+[A, φ(B)]ξ⁎ for all A, B ∈ �� if and only if φ is an additive ⁎-derivation and φ(ξA) = ξφ(A) for all A ∈ ��.

THE GENERALIZED NORMAL STATE SPACE AND UNITAL NORMAL COMPLETELY POSITIVE MAP

  • Sa Ge Lee
    • Journal of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.237-257
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    • 1998
  • By introducing the notion of a generalized normal state space, we give a necessary and sufficient condition for that there exists a unital normal completely map from a von Neumann algebra into another, in terms of their generalized normal state spaces.

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NONCOMMUTATIVE CONTINUOUS FUNCTIONS

  • Don, Hadwin;Llolsten, Kaonga;Ben, Mathes
    • Journal of the Korean Mathematical Society
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    • v.40 no.5
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    • pp.789-830
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    • 2003
  • By forming completions of families of noncommutative polynomials, we define a notion of noncommutative continuous function and locally bounded Borel function that give a noncommutative analogue of the functional calculus for elements of commutative $C^{*}$-algebras and von Neumann algebras. These notions give a precise meaning to $C^{*}$-algebras defined by generator and relations and we show how they relate to many parts of operator and operator algebra theory.