2007 Fiscal Year Final Research Report Summary
Entropy in noncommutative dynamical systems, structure of operator algebras and groups with amenable actions
| Project/Area Number |
17540193
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka Kyoiku University |
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Principal Investigator |
CHODA Marie Osaka Kyoiku University, Emeritus professor (80030378)
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| 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)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Operator Algebars / Automorphism / Amenable / Non-commutative dynamical system / Entropy / State / Relative Entropy / Maximal abelian subalgebra |
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| 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.
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Research Products
(14 results)
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[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
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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
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