Introduction to von Neumann algebras, Lecture 4 (group von Neumann algebras)
by Orr Shalit
As the main reference for this lecture we use (more-or-less) Section 1.3 in the notes by Anantharaman and Popa (here is a link to the notes on Popa’s homepage).
As for exercises:
Exercise A: Prove that has the ICC property.
Exercise B: Prove that there is an increasing sequence of von Neumann subalgebras of , such that is *-isomorphic to and such that .
Exercise C: Prove that the free group () has the ICC property.
Exercise D: Prove that . What can you say about ? (May require more advanced material: What can you say about , where is a countable discrete abelian group?).
Exercise E: We will later see that is not isomorphic to . It might be a nice exercise to think about it now (it might also be not a nice exercise, take your chances).
Exercise F: Let be a left convolver, and let be the corresponding convolution operator. Find the adjoint .
Exercise G: Prove that is a commutative group, if and only if (or ) is commutative, and that this happens if and only if .
Exercise H: Prove that (where is the usual trace) is the unique linear functional on that satisfies and for all .