Harmonic Analysis on Operator Algebras
| Project/Area Number |
16540190
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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 | Kyoto University |
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Principal Investigator |
IZUMI Masaki Kyoto University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (80232362)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Operator Algebras / Poisson Boundary / C^*-Algebras / von Neumann Algebras / 双曲群 / KMS状態 / von Neumann環 / Poisson boundary / quantum groups / von Neumann algebras / C^*-algebras / quantum probability |
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| Research Abstract |
For various models appearing quantum group actions and KMS states for the gauge action of the Cuntz-Pimsner algebra, I determined the (non-commutative) Poisson boundaries. In particular, I determined the flow of weights introduced by Connes and Takesaki for several models. Since the definition of the flow of weights involves ergodic decomposition, it is not so easy to give a concrete description in general.
The set of bounded operators B(H) of a Hilbert space H is a von Neumann algebra. A one-parameter semigroup of unit preserving endomorphism of B(H) is said to be an E_0-semigroup. E_0-semigroups are classified into three categories, type I, type II, and type III, and except for the type I case, the structure of E_0-semigroups is not well-understood. With R. Srinivasan, I constructed uncountably many mutually non-cocycle conjugate E_O-semigroups of type III. Before our construction, the only known such examples were constructed by Tsirelson. Since Tsirelson's invariant is trivial for our examples, his method can not distinguish our examples. For a given E_0-semigroup and for an open subset U of the unit interval [0,1], one can associate a von Neumann algebra A(U), which is a cocycle conjugacy invariant of the E_0-semigroup. Murray and von Numann classified von Neumann algebras into three categories, type I, type II, and type III, which is nothing to do with the type classification of E_0-semigroups a priori. For type I and type II E_0-semigroups, the von Neumann algebra A(U) is always of type I. We show that for our examples, the von Neumann algebra A(U) may be of type III according to the shape of the set U.
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Report
(4 results)
Research Products
(9 results)
