Group actions on C*-algebras and their crossed products

摘要Phillips 首席研究员 N. Christopher Phillips 提议跟进他最近与 Qing Lin 在最小微分同胚交叉积方面的合作，以及他最近在康托集上有限生成的自由交换群的自由最小作用交叉积方面的结果。这些结果，特别是证明方法，表明更强的定理应该成立。具体来说，考虑可数服从群在具有有限覆盖维数的紧凑度量空间上的最小且本质上自由的动作。最终目标是证明这种作用的变换群 C* 代数是递归次齐次 C* 代数的直接极限，没有维度增长。特别是，它应该具有稳定的排名一、真实的排名零或一以及预测的取消。虽然上述猜想还远未得到证实，但首席研究员希望通过对各个方面进行孤立分析，取得实质性进展；我们的想法是随后将各个部分组合在一起。首席研究员还建议在适当的情况下研究此类代数的光滑对应物，研究所得 C* 代数的同构分类，并研究与拓扑动力学中轨道等效问题的联系。动力学系统由一个空间和满足适当数学条件的该空间的变换集合组成。 举个例子，考虑物理系统的一组可能状态及其时间演变：转换指定给定时间和初始状态在经过这么长时间后系统将处于什么状态。另一个例子是物理空间及其基本对称性，例如狭义相对论中作用于时空的洛伦兹群。如果空间与变换之间的关系简单，则可以直接研究动力系统。当这种关系很复杂时，引入附加对象通常很有用；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，有时物理感兴趣的物体与它的关系比与原始动力系统的关系更密切。一个例子是电子在准晶体中移动的薛定谔算符。该项目的目的是了解动力系统复杂但 C* 代数似乎易于分析的情况下的变换群 C* 代数。 （这包括准晶体情况。）它还旨在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* 代数相关的对象，但保留有关原始动力学的更多信息。

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.