# On existence of fixed point operator

Let $$M$$ be an infinite dimensional non-type $$I$$ factor, given $$\xi$$ in $$\mathcal{H}$$, does there exist a not identify operator $$x$$ in $$M$$ such that $$x\xi=\xi$$, I have tried with taking projection $$P_{\xi}:\mathcal{H}\rightarrow [M'\xi]$$, this works unless $$P_{\xi}\neq I$$, but how to tackle the case when $$P_{\xi}=I$$.

No, there does not necessarily exist such an $$x$$. For example, if $$M$$ is a $$II_1$$ factor with trace $$\tau$$, $$\mathcal{H} = L^2(M,\tau)$$ and $$\xi = 1$$ (the identity of $$M$$, seen in $$L^2(M,\tau)$$), then $$x\xi=\xi$$ if and only if $$x=1$$.
• That means cyclic vectors for $M'$ is creating the problems – user136400 Apr 30 at 7:57