Operator Algebras and Self-Similar Actions

算子代数和自相似动作

• 批准号: RGPIN-2018-04003
• 负责人: Yang, Dilian
• 金额: $ 1.68万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669162
• 关键词: Operator; Algebras; Self; Similar; Actions

Operator algebras are algebras generated by a family of continuous linear transformations on Hilbert spaces. They are originally from quantum physics. They are now closely related to other areas, such as quantum computing, quantum information, signal processing, representation theory and differential geometry. They are playing increasingly important roles in modern mathematics.******Intuitively speaking, an object is called self-similar if its parts are similar to the whole. So self-similarities are naturally related to symmetries appearing in fractal geometry and dynamics. Usually, they are encoded by semigroups/inverse semigroups/groupoids induced from self-similarities, which are isomorphisms between the parts of given self-similar objects on different scales. Self-similar actions have been attracting a great deal of attention in the fields of geometric group theory and operator algebras.******My proposal mainly focuses on the interplay between operator algebras and self-similar actions. This has deep connections with a lot of hot and interesting topics, such as higher-rank graph algebras, self-similar actions, and the Yang-Baxter equation. To be more precise, given a higher-rank graph and a group, suppose that the group acts on the higher-rank graph self-similarly. To this self-similar action, one then can associate some important operator algebras, such as the Toeplitz type C*-algebra and the Cuntz-Pimsner-type C*-algebra. In particular, the Cuntz-Pimsner type C*-algebra, by definition, is the universal C*-algebra with the following properties: There are a Cuntz-Krieger representation of the higher-rank graph and a unitary representation of the group, such that these two representations are naturally compatible with respect to the given self-similar higher-rank graph action. Even in the case of rank-1 graph actions, those C*-algebras include many well-known and important classes, such as Katsura algebras (which exhaust all Kirchberg algebras) and Nekrashevych C*-algebras (which are first operator algebras essentially induced from self-similar actions). It is also a fact that higher-rank graph algebras are much more complicated than rank-1 graph algebras. Therefore, it would be no surprise at all that those operator algebras coming from self-similar higher-rank graph actions embrace more interesting, important classes of operator algebras, and are also much more sophisticated than those from self-similar graph actions. On one hand, the case of rank-1 graph self-similar actions will give us some intuitions and motivations to our development of the higher-rank case. On the other hand, what we will gain from working on the higher-rank case will also give us a better understanding of the rank-1 case.******My proposal is an initiative to the study of self-similar higher-rank graph actions, and will open a lot of interesting and important topics which are worth studying in the future.

操作员代数是由希尔伯特空间上的连续线性变换家族产生的代数。它们最初来自量子物理学。它们现在与其他领域密切相关，例如量子计算，量子信息，信号处理，表示理论和差异几何形状。他们在现代数学中扮演着越来越重要的角色。因此，自相似性自然与分形几何形状和动力学中出现的对称性有关。通常，它们是由由自相似性引起的半群/逆半群/群体/群体编码的，它们是不同尺度上给定的自相似对象的各个部分之间的同构。自我相似的行动在几何群体理论和操作员代数的领域吸引了很多关注。这与许多热门有趣的主题具有深厚的联系，例如高级图代数，自相似动作和杨巴克斯特方程。更确切地说，给定较高的图和组，假设该组作用于更高级别的图形。为了进行这种自相似的动作，然后可以将一些重要的操作员代数关联，例如Toeplitz型C*-Algebra和Cuntz-Pimsner-type C*-Algebra。特别地，根据定义，Cuntz-Pimsner型C*-Algebra是具有以下属性的通用C*-Algebra：具有较高级别图的Cuntz-Krieger表示，并且该组的单位表示，因此这两个表示与给定的自我效果较高的较高较高的较高范围的图形自然兼容。即使在Rank-1图形动作的情况下，这些C*-Algebras也包括许多知名和重要类别，例如Katsura代数（耗尽所有Kirchberg代数）和Nekrashevych C*-Algebras（它们是由自相似动作诱导的第一操作代数）。同样，较高的图表代数比Rank-1图代数复杂得多。因此，从自相似的高级图形动作中的那些操作员代数包含更有趣，重要的操作员代数类别也就不足为奇了，并且比自相似的图形动作更复杂。一方面，Rank-1图的情况自相似的动作将为我们提供一些直觉和动机，以使我们开发更高的案例。另一方面，我们将从较高的案例工作中获得的收益也将使我们对排名1案例有更好的了解。