Global and internal structure of operator algebras

算子代数的全局和内部结构

• 批准号: RGPIN-2015-06205
• 负责人: Marcoux, Laurent
• 金额: $ 1.02万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675405
• 关键词: Global; internal; structure; operator; algebras

My research is in an area of Pure Mathematics known as Operator Theory and Operator Algebras. Very loosely speaking, this is an area which studies sets of linear transformations, of which rotations, reflections, and dilations in our three-dimensional space R^3 are but three examples. My interest is to study linear transformations in the infinite-dimensional analogues of R^3 known as Hilbert spaces. Some of these collections of linear transformations (or operators) possess certain algebraic properties, allowing us to scale the members, to compose two transformations, or add two transformations, and yet still remain in the collection. For example, if we compose two rotations, we simply obtain another rotation. If we add two translations, we obtain another translation. Such collections are referred to as algebras. ******The first problem I propose to study asks whether or not two a priori different looking algebras are actually similar, where similar is used here in both a precise mathematical sense as well as the usual sense. That is, one may apply two different processes to a given algebra of transformations, and ask how the results of these processes are related.******One process involves applying a representation of the algebra (i.e. a particular way of describing the transformations in the algebra), and the second process involves a *-representation (i.e. a representation which respects the notion of perpendicularity of vectors in our spaces). One wishes to understand when these two processes are in some sense equivalent. This is an extremely interesting and important problem which was first proposed sixty years ago and has attracted the attention of some of the best mathematicians in the area.******A natural way to sort different classes of objects is to identify a characteristic which is inherent in each class, and to look for that characteristic in each object you examine. The mathematical equivalent of this process is referred to as finding an invariant for your class. The second problem which I propose to study is to determine the so-called "similarity degree" of a class of operator algebras. This degree is an invariant which can be used to partition various collections of operator algebras into distinct equivalence classes. This is a new tool recently developed in an attempt to solve the previous problem, and seems very promising.******The third problem I propose to study is to study the actual linear transformations within a given algebra of operators, and to characterize equivalence classes of these transformations, as well as determining which elements of the algebra can be approximated in some sense by certain special elements.  **

我的研究属于纯数学领域，称为算子理论和算子代数。非常宽松地说，这是一个研究线性变换集的领域，其中包括三维空间 R^ 中的旋转、反射和膨胀。 3 只是三个例子。我的兴趣是研究 R^3 的无限维类似物（称为希尔伯特空间）中的线性变换，其中一些线性变换（或运算符）集合具有某些代数性质，使我们能够使用这些线性变换。缩放成员，组成两个变换，或者添加两个变换，但仍然保留在集合中。例如，如果我们组成两个旋转，我们只需获得另一个旋转，如果我们添加两个平移，我们就会获得另一个平移。 ******我建议研究的第一个问题是问两个先验不同的代数实际上是否相似，这里的相似既是在精确的数学意义上也是在通常的意义上。也就是说，可以应用。对给定的变换代数进行两个不同的过程，并询问这些过程的结果如何相关。******一个过程涉及应用代数的表示（即描述代数中的变换的特定方式），第二个过程涉及 * 表示（即尊重空间中向量垂直概念的表示）。人们希望了解这两个过程在某种意义上是否等价。这是一个非常有趣且重要的问题。六十年前首次提出，并引起了该领域一些最优秀数学家的注意。******对不同类别的对象进行排序的自然方法是识别每个类别固有的特征，并查看对于您检查的每个对象的该特征，该过程的数学等效项称为找到您的类的不变量。我建议研究的第二个问题是确定一类运算符的所谓“相似度”。这个度数是一个不变量，可以是用于将各种算子代数集合划分为不同的等价类。这是最近开发的一个新工具，试图解决前面的问题，并且看起来很有前途。******我建议研究的第三个问题是研究。给定算子代数内的实际线性变换，并表征这些变换的等价类，以及确定代数的哪些元素可以在某种意义上由某些特殊元素来近似。