Study on non-commutative dynamical systems

非交换动力系统是一个非交换代数、一个群或类群对象及其对该代数的作用的三重。当代数是可交换的，我们称其为经典系统，我们一般的态度是研究经典情况，并尝试将得到的结果推广到非可交换系统，然后看看是否有适合于非可交换情况的东西。通过这种方法，我们得到了一些结果，澄清了在群紧的情况下作用、交叉积和（协变）表示之间的关系。特别是，当代数是一个简单的C^*-代数时，在许多情况下，作用将被视为准积类型。当代数为某W^*-代数，群为可服从离散群时，得到了作用的外共轭类的分类结果。当代数是C^*-代数时，在行为本身没有任何条件的情况下，不存在这样的代数。但现在我们成功地证明了某些系统的Rohlin性质，其W^*版本在分类理论中起着重要作用，并且被认为不太可能成立，并且情况正在迅速改变。我们目前正致力于在更广泛的系统中证明这一特性。这也为C^*-代数的新类别调用了分类理论，其中包括C^*-代数，如Cuntz代数。另一方面，我们现在知道，即使对于（技术上）相当简单的C^*-代数，也可能有各种类型的作用；我们可能需要新的变体来得到一个完整的答案。

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.