Amenability, structure and regularity of group actions on C*-algebras

在数学中，对对称性和群作用的研究是最基本和最多产的研究领域之一。在算子代数中，整数作用的分类在康纳斯获奖的第三类因子唯一性证明中起了重要作用。深远的概括最终导致了对顺从因素的顺从群体行为的分类。尽管努力，该地区的C*-动力学是远远低于发达国家比其冯诺依曼代数对应。然而，最近的突破为该领域提供了新的动力。该项目将在C*-动力学方面取得非凡的进展，巩固该领域作为C*-代数中最活跃的领域之一。沿着这个项目的主要目标，我们也将探索一些与C*-代数和冯诺依曼代数上的动力学相关的问题，其解决方案将提高我们对该领域的理解。其中一些问题与拓扑和可测动力学有着紧密的联系，因此具有跨学科的性质。该项目包括三个基本研究单元，目标雄心勃勃，前景广阔。在单元A中，我们将阐明群体的顺从性与其行动的余圈刚性之间的关系。我们期望顺从群体的行动会相当容易处理，这表明他们的分类应该是可能的。在单元B中进行的一个关键步骤是构造任何顺从群在Jiang-Su代数上的模型作用，类似于对超有限II-1因子所做的。这样的模式也可能在我们的环境中发挥核心作用。具体地说，模型作用的吸收预计相当于一些自然发生的游离和罗克林型性质;这将在单元C中探讨。建立这些事实将是关键，表明吸收了模型行动的行动可以被分类。我们项目的一个显着特点是冯诺依曼代数的思想和技术将在很大程度上以新颖的方式与C*-代数中的工具相结合，以在群作用的研究中取得显着进展。如今，C*-代数和冯诺依曼代数领域已经变得如此专业，只有少数研究人员在他们的交叉工作。然而，最近已经清楚的是，这两个领域都将从更密切的互动中受益。例如，核C*-代数的结构理论的许多重大进展依赖于专门在顺从因子的设置中开发的技术。也有一些例子，其中C*-代数方法已被证明是有用的冯诺依曼代数。C*-代数和冯诺依曼代数的理论是紧密交织在一起的，这个项目的目的是利用它们的相互依赖性。

In mathematics, the study of symmetries and group actions is one of the most fundamental and prolific fields of research. In operator algebras, the classification of integer actions was instrumental in Connes' award-winning proof of uniqueness of amenable type III factors. Far reaching generalizations culminated in the classification of amenable group actions on amenable factors. Despite the efforts, the area of C*-dynamics is far less developed than its von Neumann algebraic counterpart. However, recent breakthroughs have given new impetus to the field. This project will make exceptional progress in C*-dynamics, consolidating the area as one of the most active ones in C*-algebras. Along with the main goals of this project, we will also explore a number of problems related to dynamics both on C*-algebras and von Neumann algebras, whose solution will improve our understanding of the field. Some of these problems have tight connections to topological and measurable dynamics, and are thus of an interdisciplinary nature.The project contains three fundamental Research Units, with ambitious and promising goals. In Unit A, we will clarify the relationship between amenability of a group and cocycle-rigidity of its actions. We expect actions of amenable groups to be considerably more tractable, suggesting that their classification ought to be possible. A crucial step, to be carried out in Unit B, is constructing a model action of any amenable group on the Jiang-Su algebra, similarly to what was done for the hyperfinite II-1 factor. Such a model is likely to play a central role in our setting as well. Concretely, absorption of the model action is expected to be equivalent to a number of naturally occurring freeness- and Rokhlin-type properties; this will be explored in Unit C. Establishing these facts will be key in showing that actions absorbing the model action can be classified. A distinguishing feature of our project is the large extent to which ideas and techniques from von Neumann algebras will be combined in novel ways with tools in C*-algebras, to make outstanding advances in the study of group actions. Nowadays, the fields of C*-algebras and von Neumann algebras have become so specialized that only few researchers work in their intersection. However, it has recently become clear that both areas would profit from closer interactions. For instance, many significant advances in the structure theory of nuclear C*-algebras relied on techniques developed specifically in the setting of amenable factors. There are also a number of instances where C*-algebraic methods have proved to be useful in von Neumann algebras. The theories of C*-algebras and von Neumann algebras are intimately intertwined, and this project aims at exploiting their interdependence.