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

在数学中，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Connes获奖证明服从类型III因子的唯一性中起到了重要作用。影响深远的概括最终归结为对顺从因素的顺从群体行为的分类。尽管做出了这些努力，但C*动力学领域远远落后于冯·诺依曼代数的同行。然而，最近的突破给该领域带来了新的动力。这个项目将在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的地位。除了这个项目的主要目标，我们还将探索一些与C*-代数和von Neumann代数上的动力学有关的问题，这些问题的解决将提高我们对该领域的理解。其中一些问题与拓扑和可测量的动力学有密切的联系，因此具有跨学科的性质。该项目包括三个基础研究单位，具有雄心勃勃和有前途的目标。在单元A中，我们将阐明群体的顺从性与其行动的共周期刚性之间的关系。我们预计，顺从的团体的行动将更容易处理，这表明他们的分类应该是可能的。在单位B中要进行的一个关键步骤是构造任何服从群在江苏代数上的模型作用，类似于对超有限II-1因子所做的操作。这样的模式可能也会在我们的环境中发挥核心作用。具体地说，模型动作的吸收预计等同于一些自然产生的自由型和罗克林型性质；这将在单元C中探索。确立这些事实将是表明吸收模型动作的动作可以分类的关键。我们项目的一个显著特点是，来自von Neumann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突出进展。如今，C*-代数和von Neumann代数的研究已经变得如此专门化，以至于很少有研究人员在它们的交集上工作。然而，最近变得明显的是，这两个领域都将从更紧密的互动中受益。例如，核C*-代数结构理论中的许多重大进展都依赖于专门在顺从因子设置中开发的技术。也有一些例子证明C*-代数方法在von Neumann代数中是有用的。C*-代数和von Neumann代数的理论紧密地交织在一起，这个项目的目的是利用它们的相互依存关系。

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.