### Introduction to von Neumann algebras, Lecture 5 (comparison of projections and classification into types of von Neumann algebras)

In the previous lecture we discussed the group von Neumann algebras, and we saw that they can never be isomorphic to . There is something fundamentally different about these algebras, and this was manifested by the existence of a trace. von Neumann algebras with traces are special, and the existence or non-existence of a trace can be used to classify von Neumann algebras, into rather broad “types”. In this lecture we will study the theory of Murray and von Neumann on the comparison of projections and the use of this theory to classify von Neumann algebras into “types”. We will also see how traces (or generalized traces) fit in. (For preparing these notes, I used Takesaki (Vol I) and Kadison-Ringrose (Vol. II).)

Most of the time we will stick to the assumption that all Hilbert spaces appearing are separable. This will only be needed at one or two spots (can you spot them?).

In addition to “Exercises”, I will start suggesting “Projects”. These projects might require investing a significant amount of time (a student is not expected to choose more than one project).