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.

我的研究涉及称为 C* 代数（读作 see-star）的数学对象，以及它们可能被赋予的附加属性，称为群作用和实结构。C* 代数出现在一些量子模型中。物理学中的机械现象。在这些模型中，C* 代数描述了系统的可观测量，例如动量，而群体行为则描述了系统随时间或参考系其他变化的演化。C* 代数的真实结构是许多自然发生的 C* 代数具有额外的对称性。将它们考虑在内为研究 C* 代数提供了额外的工具。它们最近还被用来研究拓扑绝缘体，即物理中可以在其表面导电但不能在其内部导电的物体。我的研究项目旨在对具有真实结构、群作用或两者的 C* 代数进行分类。真实的结构和与之相适应的群体行动。所针对的分类类型不仅仅是一个列表。我们使用一组称为 K 理论的工具来构建描述相关对象的关键特征的标签。这就像收集大量汽车的用户手册，而不仅仅是车牌号列表。目标结果有两部分。第一部分称为分类定理，它告诉我们，我们的手册区分了我们的对象，因此具有相同手册的任何两个对象本质上是相同的，就像两辆完全相同型号的汽车一样。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收藏中的对象。在我们的汽车类比中，我们的手册库可能包含从未制造过的汽车的计划，或者不存在于我们的批次中。一旦我们有了这样的分类，我们就可以通过阅读他们的手册来回答有关我们的对象的问题，而不必把他们分开。