From C^*-algebras to von Neumann Algebras

We will discuss a few facts about C^*-algebras and develop some necessary details about II_1 von Neumann algebras in order to understand the famous open problem called the Connes Embedding Problem. Some of the details that will be investigated include the spectrum of a C^*-algebra, characters and their applications to ultrafilters, von Neumann algebra classification, and tracial states. If time permits, we will do a quick overview of ultrafilters and ultrapowers.