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。给出关于它在特定时间的状态的完整信息，一个人应该能够在任何以后的时间预测该状态。如果动力学是“可逆的”(例如在牛顿力学中)，人们也可以在任何以前的时间确定状态。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多长时间，而不取决于系统启动的特定时间。(如果你在秋天播下一颗种子，六个月后的结果将与你在春天播种的结果不同。然而，如果全年气候相同，那么无论你什么时候种植，六个月后的结果都是一样的。)此设置是实数组操作的一个示例：允许系统运行的一组可能的时间长度。其他团体的行动也很重要。整数组对应于以离散的步骤花费时间，这有时是合适的，并且在数学上更简单。另一些群是由系统中的对称性或完全不同的考虑因素引起的。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，但对于这些系统来说，组可能是复杂的。一个方向是理解系统和由系统构造的特定代数对象之间的关系，即所谓的交叉积。另一个方向是在一种特殊情况下(其中系统的对象集合非常复杂，但群实际上是有限的)对所有可能系统的完整描述。这个项目的最大组成部分涉及将紧致度量空间上的可数服从群的极小且本质自由的作用的性质与其交叉积的结构联系起来。我们的计划是在两个特定的方向上推进我们的知识的边界：(1)将极小同胚的平均维度与交叉积(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)当群是任意的，但空间是Cantor集(最简单的空间，最复杂的群)时，证明正则性。当单C*-代数上的作用具有迹Rokhlin性质时，也希望相关的方法能对这个作用有所帮助。下一个组成部分是有限群在纯无限简单可分核C*-代数上的逐点外作用的分类，这是比已分类的作用(具有Rokhlin性质的作用)大得多的一类作用。第三部分是研究指数不等于2的勒贝格空间上的算子代数。这是一个新领域，结果出人意料地丰富。

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.