1993 Fiscal Year Final Research Report Summary
Study on non-commutative dynamical systems
| Project/Area Number |
04452006
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Hokkaido University |
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Principal Investigator |
KISHIMOTO Akitaka Hokkaido University, Faculty of Science Professor, 理学部, 教授 (00128597)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Asao Hokkaido University, Faculty of Science Associate Professor, 理学部, 助教授 (80134807)
GIYA Yoshikazu Hokkaido University, Faculty of Science Professor, 理学部, 教授 (70144110)
HAYASHI Mikihiro Hokkaido University, Faculty of Science Professor, 理学部, 教授 (40007828)
NAKAZI Takahiko Hokkaido University, Faculty of Science Professor, 理学部, 教授 (30002174)
OKABE Yasunori Hokkaido University, Faculty of Science Professor, 理学部, 教授 (30028211)
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| Project Period (FY) |
1992 – 1993
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| Keywords | Operator algebras / Non-commutativity / Dynamical systems / Groups / Action / Crossed products / Covariant representations / Rohlin property |
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| Research Abstract |
A non-commutative dynamical system is a triple of a non-commutative algebra, a group or group-like object, and its action on the algebra. When the algebra is commutative, we call the system classical and our general attitude is to study the classical case and try to generalize the results obrained to the non-commutative systems, and then to see if there is anything proper for the non-commutative case. In this way we obrained some results which clarifies the relations between actions, crossed products and (covariant) representations in the case the group is compact. In particular the action will be regarded as of quasi-product type in many cases where the algebra is a simple C^*-algebra.
When the algebra is a certain W^*-algebra and the group is an amenable discrete group, one has a classification result for outer conjugate classes of actions. When the algebra is a C^*-algebra, there had been no such without any conditions on the actions themselves. But now we succeeded in proving a Rohlin property for certain systems, whose W^* version playd an important role in the classification theory and which had been thought unlikely to hold, and the situation is rapidly changing. We are currently working on proving this property for a wider class of systems. This has also invoked a classification theory for a new class of C^*-algebras which includes C^*-algebras like Cuntz algebras. On the other hand we now know that there could be various types of actions even for a rather (technically) simple C^*-algebras ; we may need new in variants for a complete answer.
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