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Entropy in the non-commutalrive dynamical systems and the structure of algebras

Research Project

Project/Area Number 12640209
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, Faculty of education (Mathematics), Professor, 教育学部, 教授 (80030378)

Co-Investigator(Kenkyū-buntansha) NAKAI Eiichi  Osaka Kyoiku University, Faculty of education, Assistant Professor, 教育学部, 助教授 (60259900)
FUJII Masatoshi  Osaka Kyoiku University, Faculty of education, Professor, 教育学部, 教授 (10030462)
KATAYAMA Yoshikazu  Osaka Kyoiku University, Faculty of education, Professor, 教育学部, 教授 (10093395)
SADASUE Gaku  Osaka Kyoiku University, Faculty of education, Lecturer, 教育学部, 講師 (40324884)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsOperator algebras / Automorphism / Non-commutative dynamical system / Entropy / State / C^*-algebra / Crossed product / Free product
Research Abstract

1. In the reduced free product of C^*-algebras, (A,φ) = (A_1, φ_1) * (A_2,φ_2) with respect to faithful states φ_1 and φ_2, A is purely infinite and simple if A_1 is a reduced crossed product B ×_<a,r> G for G an infinite group, if φ_1 is well behaved with respect, to this crossed product decomposition, if A_2 ≠ C and if φ is not a trace.
2. It is shown that for two dynamical approximation entropies (one C^* and one W^*) the implementing inner automorphism in a crossed product A ×_α Z has the same entropy value as the automorphism α. Using the techniques, in the proof, an example of a highly ergodic non-asymptotically abelian automorphism with topological entropy zero is also given. More specifically, it is shown that the free shifts on the Cuntz algebra Ο_∞ and the reduced free group C^*-algebra C^*_γ(F∞) have topological entropy zero.
3. We defined an entropic invariant for automorphisms on amenable groups and investigated basic properties. Related results were obtained by Broen-Germain independently at the same time.
4. We showed that the free group factor L(F_m) has a continuous family of non conjugate outer actions of GL(n,Z) for all m = 2, 3, ・・・, ∞, and give an estimation of the Connes-Stormer entropy for each automorphism appearing in the actions. By restricting them to subgroups with Kazhdan's property T (for an example SL(n, Z)), we have a continuous family of non co-cycle conjugate outer actions on L(F_m),m 【greater than or equal】 2.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Marie Choda: "Entropy of Crossed Products and Entropy of Free Products"Journal of Operator Theory. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie Choda: "Actions of the matrix groups on the free group factors and entropy of automorphisms"Proceedings of 4th Operator Algebras International Conference. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nathanial Brown, Marie Choda: "Approximation entropies in crossed products with an application to free shifts"Pacific Journal of Mathematics. 198. 331-346 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie Choda: "A C*-Dynamical Entropy and Applications to Canonical Endomorphisms"Journal of Functional Analysis. 173. 453-480 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie Choda, Kenneth Dykema: "Purely Infinite, Simple C*-Algebras Arizing From Free Produc t Constructions III"Proceedings of American Mathematical Soceity. 128. 3269-3273 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie Choda: "Entropy of Cuntz's canonical endomorphism, Pacific Journal of Mathematics"Pacific Journal of Mathematics. 190. 235-245 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie CHODA: "A C^*-dynamical entropy and applications to canonical endomorphisms"Journal of Functional Analysis. Vol.173, no.2. 453-480 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie CHODA and Keneth DYKEMA: "Purely infinite, simple C^*-algebras arising from free product constructions III"Proceedings of the American Mathematical Society. Vol.128, no.11. 3269-3273 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Nathanial BROWN and Marie CHODA: "Approximation entropies in crossed products with an application to free shifts"Pacific Journal of Mathematics. Vol. 198, no.2. 331-346 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie CHODA: "Entropy of Crossed Products and Entropy of Free Products"Journal of Operator Theory. (in print).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie CHODA: "Actions of the matrix groups on the free group factors and entropy of automorphisms"Proceedings of 4th Operator Algebras International Conference (OPERATOR ALGEBRAS and MATHEMATICAL PHYSICS). (in print).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yoshikazu Katayama and Masamichi Takesaki: "The structure of the automorphism group of an approximately finite dimensional factor"Mathematical physics in mathematics and physics (Siena, 2000). 249-260

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Marie CHODA: "Actions of the matrix groups on the free group factors and entropy of automorphisms"Proceedings of 4th Operator Algebras International Conference (Operator algebras and Math. Physics). (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Marie CHODA: "Entropy of Crossed Products and Entropy of Free Products"Journal of Operator Theory. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Brown, M.CHODA: "Approximation entropies in crossed products with an application to free shifts."Pacific Journal of Mathematics. 198・2. 331-346 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.CHODA, K.Dykema: "Purely Infinite, Simple C*-Algebras Arizing From Free Product Constructions III"Proceedings of the American Mathematical Soceiety. 128・11. 3269-3273 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Marie CHODA: "A C*-Dynamical Entropy and Applications to Canonical Endomorphisms"Journal of Functional Analysis. 173・2. 453-480 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Katayama, M.Takesaki: "The structure of the automorphism group of an approximately finite dimensional factor"Mathematical phyiscs in mathematics and physics. 249-260 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Nathanial P.Brown: "Approximation entropies in crossed products with an application to free shifts"Pacific Journal of Mathematics. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Marie CHODA: "Entropy of Crossed Products and Entropy of Free Products"Journal of Operator Theory. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Marie CHODA: "A C*-dynamical entropy and applications to canonical endomorphisms"Journal of Functional Analysis. 173. 453-480 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Marie CHODA: "Purely infinite, simple C*-algebras arising from free product constructions III"Proceedings of American Mathematical Soceity. 128・11. 3269-3278 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Marie CHODA: "Entropy of Cuntz's Canonical Endmorphisms"Pacific Journal of Mathematics. 190・2. 235-245 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] Marie CHODA: "Reduced C*-crossed products by free shifts"Ergodic Theory & Dynamical Systems. 15・5. 1075-1096 (1998)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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