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 $A,B \in M_n(\mathbb{C})$ .