The structure of transformation group C*-algebras

变换群C*-代数的结构

• 批准号: 0302401
• 负责人: Norman Phillips
• 金额: --
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0302401&HistoricalAwards=false
• 关键词: structure; transformation; group; algebras; structure; transformation; group; algebras

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.

AbstrackPhillips首席研究员N. Christopher Phillips提议跟进他最近通过最小的差异性来对林林的联合合作，并通过他最近在Cantor套装上有限生成的免费Abelian群体的自由最小生成的免费Abelian群体对交叉产品的结果进行跟进。这些结果，尤其是证明方法，表明应该提出更强的定理。具体而言，请考虑一个可计数的群体在紧凑的度量空间上具有最低限度的自由作用，并具有有限的覆盖尺寸。最终的目标是证明这种作用的转换组C*代数是递归下均质C* - 代数的直接极限，没有尺寸生长。特别是，它应该具有稳定的等级，实际等级零或一个，并取消预测。尽管所述的猜想尚未得到证明，但主要研究者希望通过彼此孤立地分析其各个方面来取得重大进展。这个想法是之后将碎片放在一起。主要研究者还建议在适当的情况下调查此类代数的平稳对应物，以研究所得的C* - 代数的同构分类，并研究与拓扑动态中的轨道等效问题的联系。空间的动力学系统和满足适用数学条件的这种空间的转换。 例如，考虑物理系统的可能状态及其时间演变的集合：在给定时间和初始状态中指定的转换在这么长时间之后将进入哪个系统。另一个例子是一个物理空间及其基础对称性，例如在特殊相对论上作用于时空的洛伦兹组。如果空间和转换之间的关系很简单，则可以直接研究动态系统。当这种关系复杂时，引入其他对象通常很有用。这种自然的对象是转换组C* - 代数。研究该代数的另一个原因是，有时与原始动力学系统相比，有时具有物理感兴趣的对象更加紧密相关。一个例子是施罗辛格在准晶体中移动的电子的操作员。该项目的目的是了解动态系统复杂的情况下的转换组C* - 代数，但是C*-Algebra似乎很可能可以接受分析。 （这包含了准晶体。）它还试图更好地理解动态系统与C*-Algebra之间的关系，并在适当的情况下开始分析与C*-ergebra相关的对象，但保留有关原始动力学的更多信息。