2002 Fiscal Year Final Research Report Summary
Study of approximately transitive transformations and its application to operator algebra
| Project/Area Number |
12640214
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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 | Kyushu University |
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Principal Investigator |
HAMACHI Toshihiro Graduate School of Mathematics, Kyushu University, Prof., 大学院・数理学研究院, 教授 (20037253)
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| Co-Investigator(Kenkyū-buntansha) |
NAKADA Hitoshi Faculty of Engineering and Science, Keio University, Prof., 理工学部, 教授 (40118980)
WATATANI Yasuo Graduate School of Mathematics, Kyushu University, Prof., 大学院・数理学研究院, 教授 (00175077)
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| Project Period (FY) |
2000 – 2002
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| Keywords | ergodic transformation / von Neumann algebra / Bratteli diagram / type III-0 / symbolic dynamics |
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| Research Abstract |
A beautiful and classical result of Connes and Krieger in the theory of type III-0 injective factors that the isomorphic classes of factors are completely determined by their associated flows and also by orbit equivalence classes of ergodic nonsingular transformations. Moreover Connes and Woods used this theory to show that approximately transitive flows correspond to ITPFI factors. Since it is known that every nonsingular transformation is realized by an odometer acting on an infinite product space, it is interesting to investigate measures on the infinite product space under which the odometer action is not orbit-equivalent to any product odometer action that provides an ITPFI factor. However an explicit construction of such a measure does not exist in the literature. This is in some sense surprising, since we have shown that every non-singular transformation is orbit equivalent to a Markov odometer on a Bratteli diagram. We also gave an explicit construction of such a Markov measure. Relating to this research, the theorems about some embedding in symbolic dynamical systems were obtained (1. Embedding of shifts of finite type into the Dyck shift. (With K. Inoue) 2. Subsystems of finite type and semigroup invariants of subshifts. (With K. Inoue and W. Krieger)
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Research Products
(10 results)
