Global and internal structure of operator algebras

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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.