Global and internal structure of operator algebras

我的研究是在纯数学领域被称为算子理论和算子代数。 非常宽泛地说，这是一个研究线性变换集合的领域，三维空间R^3中的旋转、反射和膨胀只是其中的三个例子。 我的兴趣是研究R ^3的无限维类似物（即希尔伯特空间）中的线性变换。 其中一些线性变换（或运算符）的集合具有某些代数性质，允许我们缩放成员，组合两个变换，或添加两个变换，但仍然保留在集合中。 例如，如果我们合成两个旋转，我们只需获得另一个旋转。 如果我们添加两个翻译，我们会得到另一个翻译。 这样的集合称为代数。** 我打算研究的第一个问题是，两个先验不同的代数是否真的相似，这里的相似既有精确的数学意义，也有通常的意义。 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（即描述代数中变换的特定方式），第二个过程涉及 *-表示（即尊重我们空间中向量的对称性概念的表示）。 人们希望了解这两个过程在某种意义上是等价的。 这是一个极其有趣且重要的问题，六十年前首次提出，并吸引了该领域一些最好的数学家的注意。*对不同类别的对象进行分类的一种自然方法是识别每个类别中固有的特征，并在您检查的每个对象中查找该特征。 这个过程的数学等价物被称为为你的类找到一个不变量。 第二个问题，我建议研究的是，以确定所谓的“相似度”的一类算子代数。 这个度是一个不变量，它可以用来划分不同的算子代数集合为不同的等价类。 这是最近开发的一个新工具，试图解决以前的问题，似乎很有前途。第三个问题，我建议研究的是研究实际的线性变换内的一个给定的代数的运营商，并表征等价类的这些转换，以及确定哪些元素的代数可以近似在某种意义上由某些特殊的元素。 **

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.  **