Study on non-commutative dynamical systems

非交换动力系统研究

基本信息

批准号：
04452006
负责人：
KISHIMOTO Akitaka
金额：
$ 4.35万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1992
资助国家：
日本
起止时间：
1992 至 1993
项目状态：
已结题

项目摘要

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.

非共同的动力系统是非交通式代数，组或类似组的对象的三倍，及其对代数的作用。当代数是可交换的时，我们称该系统经典，我们的一般态度是研究经典案例，并试图将符合的结果概括为非交通性系统，然后看看是否有任何适当的案例。通过这种方式，我们遵守了一些结果，这些结果阐明了该组紧凑的动作，交叉产品和（协变）表示之间的关系。在许多情况下，代数是一个简单的c^* - 代数。当代数为某个w^* - 代数时，并且该组是一个可amenable的离散组时，该动作将被视为准生产类型。当代数是c^* - 代数时，就没有任何行动本身的条件。但是现在，我们成功地证明了某些系统的Rohlin属性，其W^*版本在分类理论中起着重要的作用，并且认为不可能拥有，这种情况正在迅速变化。我们目前正在为更广泛的系统提供此属性。这也调用了一个新类C^* - 代数的分类理论，其中包括C^* - 像Cuntz代数这样的代数。另一方面，我们现在知道，即使是（技术上）简单的C^* - 代数也可能采取各种类型的动作。我们可能需要新的变体才能得到完整的答案。