Structure of crossed products by amenable groups and classification of group actions

这个项目的主要焦点是动力系统。也许最重要的例子是物理系统的时间演化。给定在某一特定时间的完整状态信息，人们应该能够预测以后任何时间的状态。如果动力学是“可逆的”（如在牛顿力学中），人们也可以确定任何以前时间的状态。如果系统没有外部影响，则新状态仅取决于开始状态和允许系统运行的时间，而不取决于系统启动的特定时间。如果你在秋天种下一颗种子，六个月后的结果将与你在春天种下的结果不同。然而，如果全年气候都是一样的，那么无论你什么时候种植，六个月后的结果都是一样的。这个设置是实数组动作的一个示例：允许系统运行的可能时间长度的集合。其他团体的行动也很重要。这组整数对应于在离散的步骤中花费时间，这有时是合适的，并且在数学上更简单。其他群则是由系统的对称性或完全不同的考虑而产生的。这个项目的一个主要目标是更好地理解某些类型的动力系统，在这个阶段主要是离散的系统，但对于这些系统，群体可能是复杂的。一个方向是理解系统和一个特定代数对象之间的关系，即由系统构造的所谓交叉积。另一个方向是在特殊情况下（系统对象的集合非常复杂，但群体实际上是有限的）对所有可能系统的完整描述。这个项目的最大组成部分是将紧度量空间上可数可调群的最小和本质上自由作用的性质与其交叉积的结构联系起来。计划是在两个特定的方向上推进我们的知识边界：(1)将最小同纯的平均维数与交叉积的比较半径（最简单的群，即整数和最复杂的空间）联系起来；(2)证明当群是任意的，但空间是康托集（最简单的空间，最复杂的群）时的正则性。同时也希望相关的方法能够对具有迹迹Rokhlin性质的简单C*-代数上的作用有所帮助。下一个部分是对纯无限简单可分核C*-代数上有限群的点向外作用的分类，这是比已经分类的作用（具有Rokhlin性质的作用）大得多的一类作用。第三部分是对指数不同于2的Lebesgue空间上算子代数的研究。这是一个新领域，结果却出乎意料地丰富。

The main focus of this project is on dynamical systems. Probably the most important example is the time evolution of a physical system. Given complete information about its state at a particular time, one should be able to predict the state at any later time. If the dynamics is "reversible" (such as in Newtonian mechanics), one can also determine the state at any previous time. If there are no outside influences on the system, the new state depends only on the beginning state and how long the system was allowed to run, not on the particular time at which it started. (If you plant a seed in the fall, the result six months later will be different from the result if you plant it in the spring. However, if the climate were the same year round, the result six months later would be the same regardless of when you plant.) This setup is an example of an action of the group of real numbers: the set of possible lengths of time the system is allowed to run. Actions of other groups are also important. The group of integers corresponds to taking time in discrete steps, which is sometimes appropriate and is mathematically simpler. Other groups arise from symmetry in a system or from entirely different considerations. One main goal of this project is to better understand certain kinds of dynamical systems, at this stage primarily discrete ones but for which the groups can be otherwise complicated. One direction is to understand the relation between the system and a particular algebraic object, the so-called crossed product, constructed from the system. Another direction is the complete description in a special case (in which the collection of objects of the system is very complicated but the group is actually finite) of all possible systems.The largest component of this project involves relating the properties of a minimal and essentially free action of a countable amenable group on a compact metric space to the structure of its crossed product. The plan is to push forward the boundaries of our knowledge in two two particular directions: (1) relating the mean dimension of a minimal homeomorphism to the radius of comparison of the crossed product (simplest group, namely, the integers, and most complicated space) and (2) proving regularity properties when the group is arbitrary but the space is the Cantor set (simplest space, most complicated group). It is also hoped that related methods will help with actions on simple C*-algebras when the action has the tracial Rokhlin property. The next component is the classification of pointwise outer actions of finite groups on purely infinite simple separable nuclear C*-algebras, a much larger class of actions than the ones already classified (those with the Rokhlin property). The third component is the study of operator algebras on Lebesgue spaces with exponent different from two. This is a new field which turns out to be unexpectedly rich.