Group actions on C*-algebras and their crossed products

C* 代数及其交叉积的群作用

• 批准号: 0701076
• 负责人: Norman Phillips
• 金额: $ 18万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701076&HistoricalAwards=false
• 关键词: Group; actions; algebras; their; crossed

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：首席研究员N. Christopher Phillips提议继续他与Qing Lin最近关于极小微分同构的交叉积的研究，以及他最近关于有限生成的自由阿贝尔群在Cantor集合上的自由极小作用的交叉积的研究。这些结果，特别是证明的方法，表明更强的定理应该成立。具体来说，考虑一个可数可调群在有限覆盖维的紧度量空间上的极小且基本自由的作用。最终目的是证明这种作用的变换群C*-代数是递推次齐次C*-代数的直接极限，且无维数增长。特别是，它应该具有稳定秩为1，实数秩为0或1，并且投影可以抵消。虽然所述的猜想仍远未得到证实，但首席研究员希望通过相互孤立地分析其各个方面来取得实质性进展；我们的想法是在之后把碎片拼凑在一起。在适当的情况下，研究这些代数的光滑对应物，研究由此产生的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满足适当数学条件的空间变换集合组成。例如，考虑一个物理系统的可能状态集及其时间演化：转换指定给定的时间和初始状态，在经过那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潜在的对称性，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很简单，就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，有时物理上感兴趣的对象与它的关系比与原始动力系统的关系更密切；一个例子是准晶体中电子运动的薛定谔算符。这个项目的目的是理解变换群C*-代数在动力系统是复杂的情况下，但在C*-代数似乎可能适用于分析。（这包括准晶体外壳。）它还试图更好地理解动力系统和C*-代数之间的关系，并开始分析，在适当的情况下，与C*-代数相关的对象，但保留了更多关于原始动力学的信息。

项目成果

K-Theory and the Universal Coefficient Theorem for Simple Separable Exact C*-Algebras Not Isomorphic to Their Opposites

K-理论和简单可分精确C*-代数与逆元不同构的通用系数定理

• DOI: 10.1093/imrn/rnac358
• 发表时间: 2023
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Phillips, N Christopher;Viola, Maria Grazia
• 通讯作者: Viola, Maria Grazia