I'm learning the basics of von Neumann algebras. Every reference on the subject I can find turns to the study of projections, introduces factors and the type classification immediately after having only barely introduced what a von Neumann algebra is.
This makes it hard to follow the references since I have no idea of what they are actually trying to do/accomplish by introducing these things.
What I'm really asking is the following:
Having introduced the definition of von Neumann algebras and the bicommutant Theorem, what are open questions one tries to solve immediately and leads to the study of projections, factors and type classification?
Examples of von Neumann algebras which have interesting properties and help to solve my question are certainly welcome.
Thanks in advance.