Let $M$ be a non-amenable factor. Suppose $\alpha$ is an action of a group $\Bbb R$ on $M$.

Define $$M^{\alpha}:=\{x\in M: \alpha_t(x)=x \text{ for any} \; t\in \Bbb R\}.$$

If we know that there exits $t>0$ such that $\alpha_t=\mathrm{id}$, can we have $M^{\alpha}=\Bbb C 1 $?