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

该项目的主要重点是动力系统。也许最重要的例子是物理系统的时间演化。如果给出关于它在特定时间的状态的完整信息，人们应该能够预测在任何以后的时间的状态。如果动力学是“可逆的”（例如在牛顿力学中），人们也可以确定任何先前时间的状态。如果系统没有外部影响，新的状态只取决于开始状态和允许系统运行的时间，而不取决于它开始的特定时间。(If你在秋天种下种子，六个月后的结果会与春天种下的结果不同。然而，如果全年气候相同，那么无论你何时种植，六个月后的结果都是一样的。这个设置是一个真实的数字组的动作的例子：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。整数组对应于以离散步骤计算时间，这有时是合适的，并且在数学上更简单。另一些群则是由于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，但对于这些群体来说可能是复杂的。一个方向是理解系统和一个特定的代数对象之间的关系，所谓的交叉积，从系统构造。另一个方向是在一个特殊的情况下（系统的对象的集合是非常复杂的，但该集团实际上是有限的）所有可能的systems.The最大的组成部分，这个项目涉及到的性质的最小和基本自由的行动可数顺从组在一个紧凑的度量空间的结构，其交叉产品。我们的计划是在两个特定的方向上推进我们的知识边界：（1）将最小同胚的平均维数与交叉积（最简单的群，即整数和最复杂的空间）的比较半径联系起来;（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对具有迹Rokhlin性质的单C*-代数上的作用的研究有所帮助。下一个组成部分是分类有限群在纯无限单可分核C*-代数上的逐点外作用，这是一个比已经分类的作用大得多的作用类（具有Rokhlin性质的那些）。第三部分是研究指数不等于2的Lebesgue空间上的算子代数。这是一个新的领域，结果出乎意料地丰富。

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.