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 nonidentity 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?
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1$\begingroup$ 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 $M$). $\endgroup$ – Matthew Daws Sep 12 at 9:24

$\begingroup$ 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. $\endgroup$ – YCor Sep 12 at 9:50

2$\begingroup$ 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. $\endgroup$ – Narutaka OZAWA Sep 13 at 0:48

$\begingroup$ @Ozawa Sir, can you please elaborate the reason why this example will work? $\endgroup$ – user136400 Sep 13 at 6:54