Let $\mathbb{Z}$ be the additive group of the integers. What is the Von Neumann algebra of $\mathbb{Z}$?
Let $M$ be the Von Neumann algebra of $\mathbb{Z}$.
I know that we have $M \subset B(\ell^2(\mathbb{Z}))$. But what are the linear maps in $M$ here?
According to the definition of a Von Neumann algebra, we have that if $a \in M$, then $a^* \in M$, $M = M''$.
But I don't really see what the Von Neumann algebra of $\mathbb{Z}$ is like.
Thank you in advance!
If $\Gamma$ is a discrete group (in your case $\Gamma = \mathbb{Z}$), you can consider the injective $*$-homomorphism $$\lambda: \mathbb{C}[\Gamma]\to B(\ell^2(\Gamma))$$ defined by $\lambda(s)\delta_t = \delta_{st}$ for $s,t \in \Gamma$. Then the von Neumann algebra of $\Gamma$ is exactly the set $$L(\Gamma):= \lambda(\mathbb{C}[\Gamma])'' \subseteq B(\ell^2(\Gamma)).$$
In your case, $\lambda(m)\delta_n = \delta_{m+n}$ for $m,n \in \mathbb{Z}$.
Here is an explicit description of $M$ which does not refer to the representation $\lambda$. They are the operators that are "constant down the diagonals" (think of $\ell^2$-operators as infinite matrices to see where the terminology comes from): $$L(\Gamma) = \{T \in B(\ell^2(\Gamma))\mid \forall s,t,x,y \in \Gamma: (ts^{-1} = yx^{-1}\implies \langle T\delta_s, \delta_t\rangle = \langle T \delta_x, \delta_y\rangle)\}$$
You can find all this stuff on p43-44 of the book "$C^*$-algebras and finite-dimensional approximations" by Brown and Ozawa.