# On examples of action of C-star simple group on von Neumann algebra

Can there exist a faithful action of a $$C^{*}$$-simple group $$G$$ on a von Neumann algebra $$(M,\varphi)$$ equipped with faithful normal state $$\varphi$$ such that action preserves the state $$\varphi$$ and the identity automorphism can be approximated by non-identity automorphisms i.e, there exists $$g_{n}$$ $$\in$$ $$G$$, such that \begin{align*} \|\alpha_{g_{n}}(x)-x\|_{2}\rightarrow 0, \end{align*} as $$n\rightarrow \infty$$ $$\forall$$ $$x$$ $$\in$$ $$M$$ and $$\alpha_{g_{n}}\neq I$$, $$\|\cdot\|_{2}$$ is the GNS Hilbert space $$L^{2}(M,\varphi)$$ norm?

• Some questions which might improve the question: what is a $C^*$-simple group? What is $\|\cdot\|_2$ (asked because you have placed no restriction on your von Neumann algebra $M$). – Matthew Daws Sep 12 at 9:24
• A group is said to be $C^*$-simple if its reduced $C^*$-algebra is simple (every proper quotient is reduced to $\{0\}$). This also means that whenever a nonzero unitary representation is weakly contained in the regular representation, then it weakly contains the regular representation. – YCor Sep 12 at 9:50
• It's obviously possible for $G=F_\infty$. Also, e.g., if $G$ embeds densely into a compact group $K$, then $G\curvearrowright L^\infty(K)$ is an example. – Narutaka OZAWA Sep 13 at 0:48
• @Ozawa Sir, can you please elaborate the reason why this example will work? – user136400 Sep 13 at 6:54