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Entropy in noncommutative dynamical systems, structure of operator algebras and groups with amenable actions

Research Project

Project/Area Number 17540193
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOsaka Kyoiku University

Principal Investigator

CHODA Marie  Osaka Kyoiku University, Emeritus professor (80030378)

Co-Investigator(Kenkyū-buntansha) KATAYAMA Yoshikazu  Osaka Kyoiku University, Fucalty of Education, Professor (10093395)
FUJII Masatoshi  Osaka Kyoiku University, Furalty of Education, Professor (10030462)
OKAYASU Rui  Osaka Kyoiku University, Fucalty of Education, Associate Professor (70362746)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,140,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsOperator Algebars / Automorphism / Amenable / Non-commutative dynamical system / Entropy / State / Relative Entropy / Maximal abelian subalgebra / 群 / 状態
Research Abstract

(1) Let $\sigma $ be the automorphism of the free group $F_\infty$ which is arising from a permutation of the free generators of $F_\infty.$ The $\sigma $ naturally induces the automorphism $\hat \sigma $ of the reduced $C^*$-algebra $C^* r(F_\infty),$ and also the automorphism $\bar{\hat \sigma} $ of the group factor $L(F_\infty). $
We showed that the Brown-Germain entropy $ha(\sigma )$ is zero. This implies that the Brown-Voiculescu topological entropy $ht(\hat \sigma ), $ the Connes-Narnhofer-Thirring dynamical entropy $h_\phi (\hat \sigma )$ and the Connes-St\o rmer entropy $H(\bar{\hat \sigma} )$ are all zero.
(2) Let $A$ and $B$ be maximal abelian *-subalgebras of the $n\times n$ complex matrices.
we modify the Connes-St$\o$rmer relative entropy and the Connes relative entropy with respect to a state $\phi,$ and introduce the two kinds of the constant $h(A | B)$ and $h_\phi(A | B).$ For the unistochastic matrix $b(u)$ defined by a unitary $u$ with $B = uAu^*,$ we show that $h(A | B)$ is the entropy $H (b(u)) $ of $b(u).$ This is obtained by our computation of $h_\phi (A | B). $ The $h(A I B)$ attains to the maximal value $\log n$ if and only if the pair $\{A, B\}$ is orthogonal in the sense of Popa.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (28 results)

All 2008 2007 2006 2005 Other

All Journal Article (20 results) (of which Peer Reviewed: 6 results) Presentation (8 results)

  • [Journal Article] Relative entropy for maximal abelian subalgebras of matrices is the entropy of unistochastic matrices2008

    • Author(s)
      Marie Choda
    • Journal Title

      International Journal of Mathematics (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The ratio set of the harmonic measure of a random walk on a hyperbolic group2008

    • Author(s)
      Izumi, M., Neshveyev, S., Okayasu, R
    • Journal Title

      Israel J. Math 163

      Pages: 285-316

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The ratio set of the harmonic measure of a random walk on a hyperbolic group2008

    • Author(s)
      Masaki Izumi
    • Journal Title

      Israel J. Math 163

      Pages: 285-316

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Outer actions of a discrete amenable group on approximately finite dimensional factors. II2007

    • Author(s)
      Yosikazu, Katayama, M. Takesaki
    • Journal Title

      Math. Scand 100-1

      Pages: 75-129

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A reverse inequality for the weighted geometric mean due to Lawson-Lim2007

    • Author(s)
      Fujii, Jun, Ichi, Fujii, Masatoshi, Nakamura, Masahiro, Pev, cari'c, Josip, Seo, Yuki
    • Journal Title

      Linear Algebra Appl 427-2-3

      Pages: 272-284

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Outer actions of a discrete amenable group on approximately finite dimensional factors. II2007

    • Author(s)
      Yoshikazu Katayama
    • Journal Title

      Math. Scand. 100

      Pages: 75-129

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Entropy for automorphisms of free groups2006

    • Author(s)
      Marie Choda
    • Journal Title

      Proceedings of the Aerican Mathematical Soceity 134

      Pages: 2905-2911

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Perturbation theoretic entropy of the boundary actions of free groups2006

    • Author(s)
      Rui Okayasu
    • Journal Title

      Proceedings of the Aerican Mathematical Soceit 134

      Pages: 1771-1776

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Entropy for automorphisms of free groups2006

    • Author(s)
      Marie Choda
    • Journal Title

      Proceedings of the Aerican Mathematical Soceity 134・10

      Pages: 2905-2911

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Kadison's Schwarz inequality and noncommutative Kantorovich ineauality2006

    • Author(s)
      Masatoshi Fujii, Masahiro Nakamura
    • Journal Title

      Sci. Math. Jpn 63・1

      Pages: 101-102

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Gromov hyoerbolic groups and the Macaev norm2006

    • Author(s)
      Rui Okayasu
    • Journal Title

      Pacific Journal of Mathematics 223・1

      Pages: 141-157

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Perturnation theoretic entropy of the boundary actions of free groups2006

    • Author(s)
      Rui Okayasu
    • Journal Title

      Proceedings of the Aerican Mathematical Soceity 143・6

      Pages: 1771-1776

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Entropy for automorphisms of free groups2006

    • Author(s)
      Marie Choda
    • Journal Title

      Proceecings of the American Mathematical Soceity (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Perturbation theoretic entropy of the boundary actions of free groups2006

    • Author(s)
      Rui Okayasu
    • Journal Title

      Proceecings of the American Mathematical Soceity 134・6

      Pages: 1771-1776

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An analysis on internal structure of the celebrated Furuta inequality via operator mean2005

    • Author(s)
      M.Fujii, E.Knamei, R.Nakamoto
    • Journal Title

      Sci.Math.Jpn. 62・3

      Pages: 421-427

    • Related Report
      2005 Annual Research Report
  • [Journal Article] C*-algebras associated with the fundamental graphs of groups2005

    • Author(s)
      Rui Okayasu
    • Journal Title

      Mathematica Scandinavica 97・1

      Pages: 49-72

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Entropy for automorphisms on operator algebras and related notions of entropy2005

    • Author(s)
      長田 まりゑ
    • Journal Title

      Proceeaings of the workshop on Operator Theory and Operator Algebras

      Pages: 53-63

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Classification Program - An Example-2005

    • Author(s)
      片山 良一
    • Journal Title

      Proceeaings of the workshop on Operator Theory and Operator Algebras

      Pages: 75-84

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Relative entropy for maximal abelian subalgebras of matrices and the entropy of unistochastic matrices

    • Author(s)
      Marie, Choda
    • Journal Title

      International Journal of Mathematics (in Press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Relative entropy for maximal abelian subalgebras of matrices is the entropy of unistochastic matrices

    • Author(s)
      Marie Choda
    • Journal Title

      International Journal of Mathematics (印刷中)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] The entropy of unistochastic matrices and the relative entropy for maximal abelian subalgebras of matrice2007

    • Author(s)
      Marie CHODA
    • Organizer
      10th WORKSHOP: NON-COMMUTATIVE HARMONIC ANAL YSIS WITH APPLICATIONS TO PROBABILITY,
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2007-08-10
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The entropy of unistochastic matrices and the relative entropy for maximal abelian subalgebras of matrices2007

    • Author(s)
      Marie, CHODA
    • Organizer
      10thWORKSHOR non-commutative Harmonic analysis with applications to probability
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2007-08-10
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The entropy of unistochastic matrices and the relative entropy for maximal abelian subalgebras of matrices2007

    • Author(s)
      Marie CHODA
    • Organizer
      10th WORKSHOP: non-commutative harmonic analysis with application to probability
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2007-08-10
    • Related Report
      2007 Annual Research Report
  • [Presentation] Some example of computations for entropy in operator algebras2007

    • Author(s)
      Marie, CHODA
    • Organizer
      The seminar on Discrete Harmonic Analysis
    • Place of Presentation
      Institute of Informatic of Uniwersytet Wroclawski, Poland
    • Year and Date
      2007-08-03
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the ratio set of the harmonic measure on the Gromov boundary of a hyperbolic gorup2007

    • Author(s)
      Rui, Okayasu
    • Organizer
      Discrete groups and operator algebras
    • Place of Presentation
      RIMS in Kyoto
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Entropy and the Haagerup invariant related to groups with amenable actions2006

    • Author(s)
      Marie, CHODA
    • Organizer
      ESI Lecture
    • Place of Presentation
      ESI in Vienna, Austria
    • Year and Date
      2006-02-16
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Types of von Neumann algebras arising from boundary actions of hyperbolic groups2005

    • Author(s)
      Rui, Okayasu
    • Organizer
      Development in the theory of operator algebras
    • Place of Presentation
      RIMS in Kyoto
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Perturbation theoretic entropy of boundary actions of free groups2005

    • Author(s)
      Rui, Okayasu
    • Organizer
      Conference in Japan Mathematical Soceity
    • Place of Presentation
      Okayama University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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