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。

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.