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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711696
• 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.

算子代数是由Hilbert空间上的一族连续线性变换生成的代数。它们最初来自量子物理学。它们现在与其他领域密切相关，如量子计算、量子信息、信号处理、表示理论和微分几何。它们在现代数学中扮演着越来越重要的角色。 直观地讲，如果一个对象的各部分与整体相似，则称其为自相似。因此，自相似性自然与出现在分形几何学和动力学中的对称性有关。它们通常由自相似产生的半群/逆半群/群来编码，它们是给定自相似对象在不同尺度上的部分之间的同构。自相似行为在几何群论和算子代数领域引起了极大的关注。 我的建议主要集中在算子代数和自相似作用之间的相互作用。这与许多热点和有趣的话题有着深刻的联系，如高阶图代数、自相似作用和Yang-Baxter方程。更准确地说，给定一个较高秩图和一个群，假设该群自相似地作用于较高秩图。对于这种自相似作用，可以联系到一些重要的算子代数，如Toeplitz型C*-代数和Cuntz-Pimsner型C*-代数。特别地，根据定义，Cuntz-Pimsner型C*-代数是具有以下性质的泛C*-代数：存在高秩图的Cuntz-Krieger表示和群的酉表示，使得这两个表示相对于给定的自相似高秩图作用自然地兼容。即使在秩为1的图作用的情况下，这些C*-代数也包括许多著名且重要的类，例如Katsura代数(用尽所有Kirchberg代数)和Nekrashevych C*-代数(本质上是由自相似作用诱导的第一算子代数)。另一个事实是，高阶图代数比秩1图代数复杂得多。因此，由自相似高阶图作用得到的算子代数包含了更有趣、更重要的一类算子代数，并且比来自自相似图作用的算子代数复杂得多，这也就不足为奇了。一方面，秩1图自相似行为的情形将给我们一些直觉和动机来发展更高秩情形。另一方面，从更高级别的案件中我们将获得什么，也将使我们对级别1的案件有更好的理解。 我的建议是对自相似高阶图作用的研究的一个创举，并将开启许多有趣而重要的课题，在未来值得研究。