Global and internal structure of operator algebras

我的研究方向是纯数学领域，即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，我们三维空间R^3中的旋转、反射和膨胀只是三个例子。我的兴趣是研究R^3的无限维类比中的线性变换，即所谓的希尔伯特空间。这些线性变换(或运算符)的一些集合具有某些代数属性，允许我们缩放成员、组成两个变换或添加两个变换，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。如果我们将两个翻译相加，就会得到另一个翻译。这样的集合称为代数。 我要研究的第一个问题是，两个先验上看起来不同的代数是否实际上是相似的，在这里，相似既是在精确的数学意义上使用的，也是在通常意义上使用的。也就是说，人们可以对给定的变换代数应用两个不同的过程，并询问这些过程的结果是如何相关的。 一个过程涉及应用代数的表示法(即，描述代数中变换的特定方式)，而第二个过程涉及*表示法(即，尊重空间中向量垂直的概念的表示法)。人们希望了解这两个过程在某种意义上是等价的。这是一个非常有趣和重要的问题，它是60年前首次提出的，并引起了该领域一些最优秀的数学家的注意。 对不同类别的对象进行分类的一种自然方法是确定每个类别固有的特征，并在您检查的每个对象中寻找该特征。这个过程的数学等价物被称为为您的类寻找不变量。我要研究的第二个问题是确定一类算子代数的所谓“相似度”。这个度是一个不变量，可以用来将不同的算子代数集合划分成不同的等价类。这是最近开发的一种新工具，试图解决之前的问题，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中的实际线性变换，并刻画这些变换的等价类，以及确定代数中的哪些元素在某种意义上可以由某些特殊元素来逼近。

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.