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

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.

摘要Phillips是主要研究者，N.克里斯托弗菲利普斯，建议跟进他最近的合作工作与林青交叉产品的最小的代数同态，并在他最近的结果交叉产品的自由极小行动的代数生成的自由交换群的康托集。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，考虑有限覆盖维数紧度量空间上可数顺从群的极小本质自由作用。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*-代数的无维数增长的直接极限。特别地，它应该具有稳定的秩1、真实的秩0或1以及投影的消除。虽然上述猜想还远未得到证明，但首席研究员希望通过孤立地分析它的各个方面来取得实质性进展;这个想法是以后把这些碎片放在一起。主要研究者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，并研究拓扑动力学中与轨道等价问题的联系。动力系统由空间和满足适当数学条件的该空间的变换集合组成。 作为一个例子，考虑一个物理系统的一组可能的状态及其时间演化：变换指定了给定时间和初始状态，系统在经过那么长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，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。当这个关系很复杂时，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，有时物理上感兴趣的对象与它的关系比与原始动力学系统的关系更密切;一个例子是电子在准晶体中运动的薛定谔算符。这个项目的目的是了解变换群C*-代数的情况下，动力系统是复杂的，但其中的C*-代数似乎很可能是服从分析。(This包括准晶体情况。）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*-代数相关的对象，但保留更多关于原始动力学的信息。