Group actions on C*-algebras and their crossed products

主要研究人员N·克里斯托弗·菲利普斯建议继续他最近与林青关于极小微分同胚交叉积的工作，以及他最近关于Cantor集上有限生成自由交换群的自由极小作用的交叉积的结果。这些结果，特别是证明方法，表明更强的定理应该成立。具体地说，考虑有限覆盖维紧度量空间上可数服从群的极小且本质自由的作用。最终目的是证明这样一个作用的变换群C*-代数是递归次齐次C*-代数的一个没有维度增长的直接极限。特别是，它应该有稳定的排名一，真正的排名零或一，并取消预测。虽然上述猜想仍远未得到证实，但首席调查员希望通过相互独立地分析它的各个方面来取得实质性进展；其想法是在事后将碎片拼凑在一起。主要研究者还建议在适当的情况下研究这些代数的光滑对应，研究所得到的C*-代数的同构分类，并研究与拓扑动力学中的轨道等价问题的联系。动力系统由一个空间和该空间的一组满足适当数学条件的变换组成。例如，考虑物理系统的一组可能的状态及其时间演化：对于给定的时间和初始状态，转换指定了系统在经过那么多时间后将处于什么状态。另一个例子是物理空间及其基本对称性，比如狭义相对论中作用于时空的洛伦兹群。如果空间和变换之间的关系是简单的，则可以直接研究动力系统。当这种关系很复杂时，引入额外的对象通常是有用的；一个自然的这样的对象是变换群C*-代数。研究这个代数的另一个原因是，有时感兴趣的物理对象与它的关系比与原始动力系统更密切；一个例子是电子在准晶中运动的薛定谔算符。这个项目的目的是在动力系统很复杂的情况下理解变换群C*-代数，但在这种情况下C*-代数似乎是可以分析的。(这包括准晶情况。)它还试图更好地理解动力系统和C*-代数之间的关系，并在适当的情况下开始分析与C*-代数相关但保留了更多关于原始动力学的信息的对象。

AbstractPhillipsThe Principal Investigator, N. Christopher Phillips, proposes to follow up on his recent joint work with Qing Lin on crossed products by minimal diffeomorphisms, and on his recent results on crossed products by free minimal actions of finitely generated free abelian groups on the Cantor set. These results, and particularly the methods of proof, suggest that much stronger theorems should hold. Specifically, consider a minimal and essentially free action of a countable amenable group on a compact metric space with finite covering dimension. The ultimate goal is to prove that the transformation group C*-algebra of such an action is a direct limit, with no dimension growth, of recursive subhomogeneous C*-algebras. In particular, it should have stable rank one, real rank zero or one, and cancellation of projections. While the conjecture as stated is still far from being proved, the Principal Investigator hopes to make substantial progress by analyzing various aspects of it in isolation from each other; the idea is to put the pieces together afterwards. The Principal Investigator also proposes to investigate, where appropriate, the smooth counterparts of such algebras, to investigate the isomorphism classification of the resulting C*-algebras, and to investigate connections with orbit equivalence problems in topological dynamics.A dynamical system consists of a space and a collection of transformations of this space satisfying suitable mathematical conditions. As an example, consider the set of possible states of a physical system and its time evolution: the transformations specify for a given time and initial state what state the system will be in after that much time has passed. Another example would be a physical space and its underlying symmetries, such as the Lorentz group acting on space-time in special relativity. If the relation between the space and the transformations is simple, the dynamical system can be studied directly. When this relation is complicated, it is often useful to introduce additional objects; one natural such object is the transformation group C*-algebra. An additional reason for studying this algebra is that sometimes objects of physical interest are more closely related to it than to the original dynamical system; an example is the Schroedinger operator for an electron moving in a quasicrystal. The purpose of this project is to understand the transformation group C*-algebras in cases in which the dynamical system is complicated, but in which the C*-algebra seems likely to be amenable to analysis. (This incluse the quasicrystal case.) It also seeks to better understand the relation between the dynamical system and the C*-algebra, and to begin the analysis, in appropriate cases, of objects that are related to the C*-algebra but preserve more information about the original dynamics.