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 S_\infty has the ICC property.

Exercise B: Prove that there is an increasing sequence A_n of von Neumann subalgebras of L(S_\infty), such that A_n is *-isomorphic to L(S_n) and such that \overline{\cup_n A_n}^{SOT} = L(S_\infty).

Exercise C: Prove that the free group F_n (n \geq 2) has the ICC property.

Exercise D: Prove that L(\mathbb{Z}) \cong L^\infty(\mathbb{T}). What can you say about L(\mathbb{Z}^n)? (May require more advanced material: What can you say about L(G), where G is a countable discrete abelian group?).

Exercise E: We will later see that L(F_n) is not isomorphic to L(S_\infty). 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 f \in \ell^2(G) be a left convolver, and let L_f : \ell^2(G) \to \ell^2(G) be the corresponding convolution operator. Find the adjoint L_f^*.

Exercise G: Prove that G is a commutative group, if and only if L(G) (or R(G)) is commutative, and that this happens if and only if L(G) = R(G).

Exercise H: Prove that \frac{1}{n}Tr (where Tr is the usual trace) is the unique linear functional \varphi on M_n(\mathbb{C}) that satisfies \varphi(I_n) = 1 and \varphi(AB) = \varphi(BA) for all A,B \in M_n(\mathbb{C}).