Regarding group action on vN algebras

Question

$G$ be a discrete countable group acting on $M$ via automorphisms of $M$. Does there exist a faithful normal state on $M$ which preserves the action means $\varphi(\sigma_{g}(x))=\varphi(x)$, $\sigma:G\rightarrow Aut(M)$ is a homomorphism.

Answer 1

This is false even in the Abelian. It is said that a Borel action of $G$ in a measure space $(X,\nu)$ (preserving the measure class) is of type $\mathrm{III}$ if there is no invariant semifinite measure $\mu$. You are asking for examples of type $\mathrm{III}$ actions.

Edit: If you want $\mu$ to be finite, then there are type $\mathrm{II}_\infty$ counterexamples. Let us take $SL_2(\mathbb{Z}) \subset SL_2(\mathbb{R})$ acting on $S^1 \cong SL_2(\mathbb{R}) / AN$, where $A$ are the diagonal matrices and $N$ is the nilpotent group $$ N = \Big\{ \Big( \begin{matrix} 1 & t\\ 0 & 1 \end{matrix} \Big) : t \in \mathrm{R} \Big\}. $$ Basically, you are seeing $S^1$ as the $K$-part of the $KAN$ decomposition of $SL_2(\mathbb{R})$. $SL_2(\mathbb{R})$ is nonamenable but the action is Zimmer amenable. If there were any invariant measures the group $SL_2(\mathbb{Z})$ would be amenable. The same argument also gives (non-transitive) ergodic non-measure preserving actions when $\Gamma$, a hyperbolic group, acts on its Gromov boundary $\partial \Gamma$.