New directions in C*-algebras, Groupoids, and Inverse Semigroups

C* 代数、群形和逆半群的新方向

• 批准号: RGPIN-2021-03834
• 负责人: Starling, Charles
• 金额: $ 1.89万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742742
• 关键词: New; directions; algebras; Groupoids; Inverse

Von Neumann introduced operator algebras to express quantum mechanics in a mathematical language. C*-algebras are a tractable and rich class of operator algebras, and their study has a geometric flavour---they have been described by Connes as "noncommutative geometry". Geometry is governed by its beautiful symmetries, and my program explores the extent to which C*-algebras are governed by "partial symmetries" through structures called groupoids and inverse semigroups. A groupoid is certain type of set of partial symmetries on some geometric space. To every groupoid, one may construct a C*-algebra built from it. My long-term goal is to determine whether every C*-algebra is obtainable from partial symmetries in this way; or if this is not possible, describe the largest class of C*-algebras obtainable in this way. If all or most are obtainable in this way, it would foster new research connections between the theories of operator algebras, dynamical systems, and semigroups. In 1976, the Canadian mathematician George Elliott made a major discovery: that some classes of C*-algebras are completely determined by their K-theory group. The K-theory group of a C*-algebra is a kind of listing of its (possibly infinite-dimensional) subspaces arranged in order according to their (generalized) dimension. Expanding the classes for which Elliott's result hold has been an ongoing program ever since---the "Elliott Classification Program." Giving efficient methods for computing the K-theory groups of groupoid C*-algebras would be of great value to this program, especially if work on our long term goal shows that most C*-algebras are obtained this way. It is then a short-term goal to develop such methods, and apply them to examples of recent interest, including those arising from chaotic dynamical systems and finite automata.

冯·诺依曼引入算子代数来用数学语言表达量子力学。C ~*-代数是一类丰富而易处理的算子代数，它的研究具有几何的色彩，被Connes称为“非交换几何”。几何学是由其美丽的对称性所支配的，我的程序探索了C*-代数通过称为群胚和逆半群的结构受“部分对称性”支配的程度。广群是某些几何空间上某种类型的部分对称集合。对于每一个广群，我们都可以从它构造一个C*-代数，我的长期目标是确定是否每一个C*-代数都可以用这种方法从部分对称性得到;或者如果这是不可能的，描述用这种方法得到的最大类C*-代数。如果全部或大部分都可以通过这种方式获得，那么它将促进算子代数、动力系统和半群理论之间新的研究联系。1976年，加拿大数学家乔治·艾略特（英语：George Elliott）有一个重大发现：某些类的C*-代数完全由它们的K-理论群决定。C*-代数的K-理论群是它的（可能是无限维的）子空间按照它们的（广义）维数排列的一种列表。扩大类，艾略特的结果举行一直是一个持续的计划，因为-“艾略特分类计划。“给出计算广群C*-代数的K-理论群的有效方法对这个程序有很大的价值，特别是如果我们的长期目标表明大多数C*-代数都是这样得到的。然后，它是一个短期的目标，发展这样的方法，并将它们应用到最近感兴趣的例子，包括那些产生于混沌动力系统和有限自动机。