Project/Area Number 
04452006

Research Category 
GrantinAid for General Scientific Research (B)

Allocation Type  Singleyear Grants 
Research Field 
解析学

Research Institution  Hokkaido University 
Principal Investigator 
KISHIMOTO Akitaka Hokkaido University, Faculty of Science Professor, 理学部, 教授 (00128597)

CoInvestigator(Kenkyūbuntansha) 
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)
綿谷 安男 北海道大学, 理学部, 助教授 (00175077)

Project Period (FY) 
1992 – 1993

Project Status 
Completed (Fiscal Year 1993)

Budget Amount *help 
¥6,800,000 (Direct Cost: ¥6,800,000)
Fiscal Year 1993: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1992: ¥3,700,000 (Direct Cost: ¥3,700,000)

Keywords  Operator algebras / Noncommutativity / Dynamical systems / Groups / Action / Crossed products / Covariant representations / Rohlin property / 非可換力学系 / エルゴード性 / 自己同型写像 / 正準反交換関係 / 無限テンソル積 
Research Abstract 
A noncommutative dynamical system is a triple of a noncommutative algebra, a group or grouplike 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 noncommutative systems, and then to see if there is anything proper for the noncommutative 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 quasiproduct 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.
