Classification of Real Structures and Actions on C*-algebras

点击翻译按钮获取中文摘要

My research deals with mathematical objects called C*-algebras (read see-star), along with additional properties they may be endowed with, called group actions and real structures.******C*-algebras arise, among other places, in some models of quantum mechanical phenomena in physics. In these models, the C*-algebra describes the observable quantities of the system, such as momentum, while group actions describe the evolution of the system through time or other changes of reference frame.******A real structure on a C*-algebra is an additional symmetry that many naturally occurring C*-algebras possess. Taking them into account provides extra tools for studying the C*-algebra. They have also recently been used to study topological insulators, bodies in physics that may conduct electricity over their surfaces but not in their interiors.******My research program looks to classify C*-algebras endowed with a real structure, a group action, or both a real structure and a group action that is compatible with it. The type of classification aimed at is not simply a list. We use a set of tools called K-theory to construct labels that describe the key features of the objects in question. This is like having a collection of owner's manuals for a lot full of cars, as opposed to just a list of license plate numbers. ******The results aimed at have two parts. The first part, called the classification theorem, tells us that our manuals distinguish between our objects, so that any two objects with the same manual are essentially the same, like two cars of exactly the same model. The second part, called the range of the invariant theorem, tells us which manuals correspond to objects in our collection. In our cars analogy, our library of manuals may contain plans for cars that were never built, or are not present in our lot.******Once we have such a classification, we can answer questions about our objects by reading their manuals, rather than having to take them apart.