Classification of Real Structures and Actions on C*-algebras

C* 代数实结构和作用的分类

• 批准号: RGPIN-2018-04879
• 负责人: Dean, Andrew
• 金额: $ 1.17万
• 依托单位: Lakehead University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758379
• 关键词: Classification; Real; Structures; Actions; algebras

My research deals with mathematical objects called C*-algebras (read see-star), along with additional properties they may be endowed with, called group actions and real structures.C*-algebras arise, among other places, in some models of quantum mechanical phenomena in physics. In these models, the C*-algebra describes the observable quantities of the system, such as momentum, while group actions describe the evolution of the system through time or other changes of reference frame.A real structure on a C*-algebra is an additional symmetry that many naturally occurring C*-algebras possess. Taking them into account provides extra tools for studying the C*-algebra. They have also recently been used to study topological insulators, bodies in physics that may conduct electricity over their surfaces but not in their interiors.My research program looks to classify C*-algebras endowed with a real structure, a group action, or both a real structure and a group action that is compatible with it. The type of classification aimed at is not simply a list. We use a set of tools called K-theory to construct labels that describe the key features of the objects in question. This is like having a collection of owner's manuals for a lot full of cars, as opposed to just a list of license plate numbers. The results aimed at have two parts. The first part, called the classification theorem, tells us that our manuals distinguish between our objects, so that any two objects with the same manual are essentially the same, like two cars of exactly the same model. The second part, called the range of the invariant theorem, tells us which manuals correspond to objects in our collection. In our cars analogy, our library of manuals may contain plans for cars that were never built, or are not present in our lot.Once we have such a classification, we can answer questions about our objects by reading their manuals, rather than having to take them apart.

我的研究涉及被称为C*-代数的数学对象(参见-star)，以及它们可能被赋予的其他性质，称为群作用和实结构。C*-代数在物理中的一些量子力学现象模型中出现。在这些模型中，C*-代数描述了系统的可观测量，如动量，而群作用描述了系统随时间或其他参考系的变化而演化的过程。C*-代数上的真实结构是许多自然产生的C*-代数所具有的附加对称性。将它们考虑在内，为研究C*-代数提供了额外的工具。最近，它们还被用于研究拓扑绝缘体，即物理上可以在其表面上导电，但不能在其内部导电的物体。我的研究计划旨在对具有真实结构、群作用或既具有真实结构又具有与其相容的群作用的C*-代数进行分类。针对的分类类型不是简单的清单。我们使用一组称为K-理论的工具来构造描述所讨论对象的关键特征的标签。这就像是收集了一大堆汽车的车主手册，而不是只有一个车牌号码列表。研究成果主要分为两个部分。第一部分，称为分类定理，告诉我们，我们的手册区分我们的对象，因此任何两个具有相同手册的对象本质上是相同的，就像两辆完全相同型号的汽车。第二部分称为不变定理的范围，告诉我们哪些手册对应于我们集合中的对象。在我们的汽车类比中，我们的手册库可能包含从未制造或不存在的汽车的平面图。一旦我们有了这样的分类，我们就可以通过阅读它们的手册来回答关于我们的物体的问题，而不是把它们拆开。