• Title/Summary/Keyword: von Neumann algebras

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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.

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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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.

Injective JW-algebras

  • Jamjoom, Fatmah Backer
    • Kyungpook Mathematical Journal
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    • v.47 no.2
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    • pp.267-276
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    • 2007
  • Injective JW-algebras are defined and are characterized by the existence of projections of norm 1 onto them. The relationship between the injectivity of a JW-algebra and the injectivity of its universal enveloping von Neumann algebra is established. The Jordan analgue of Theorem 3 of [3] is proved, that is, a JC-algebra A is nuclear if and only if its second dual $A^{**}$ is injective.

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MULTIPLICATION OPERATORS ON BERGMAN SPACES OVER POLYDISKS ASSOCIATED WITH INTEGER MATRIX

  • Dan, Hui;Huang, Hansong
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.41-50
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    • 2018
  • This paper mainly considers a tuple of multiplication operators on Bergman spaces over polydisks which essentially arise from a matrix, their joint reducing subspaces and associated von Neumann algebras. It is shown that there is an interesting link of the non-triviality for such von Neumann algebras with the determinant of the matrix. A complete characterization of their abelian property is given under a more general setting.

ON ACTION OF LAU ALGEBRAS ON VON NEUMANN ALGEBRAS

  • Mohammad, Ramezanpour
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.557-570
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    • 2015
  • Let $\mathbb{G}$ be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that $\hat{\mathbb{G}}$, the dual of $\mathbb{G}$, is co-amenable if and only if there is a state $m{\in}L^{\infty}(\hat{\mathbb{G}})^*$ which is invariant under a left module action of $L^1(\mathbb{G})$ on $L^{\infty}(\hat{\mathbb{G}})^*$. This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.

NONLINEAR MAPS PRESERVING THE MIXED PRODUCT *[X ⋄ Y, Z] ON *-ALGEBRAS

  • Raof Ahmad Bhat;Abbas Hussain Shikeh;Mohammad Aslam Siddeeque
    • 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.

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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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.