The map $\xi \mapsto \omega_\xi$ from a Hilbert space to the pre-dual of a von Neumann algebra is continuous

Question

Let $M$ be a von Neumann algebra on a Hilbert space $H$. For any $\xi \in H$, we define $$\omega_\xi(x)=\langle x\xi,\xi\rangle,~~\text{ for } x \in M.$$ Then, $\omega_\xi$ defines a normal positive linear functional on $M$ and belongs to pre-dual of $M$, that is, $\omega_\xi \in M_*$ for all $\xi \in H.$ Now define a map $\tau:H \to M_*$ be $$\tau(\xi)=\omega_\xi,~~\forall ~ \xi \in H.$$ I want to show that the map $\tau$ is continuous.
So it is enough to show that $\tau^{-1}(U)$ is open in $H$ for any normed open set $U$ in $M_*$. Please help me to solve this. Thank you for your effort.

Answer 1

It suffices to find, for each $\epsilon > 0$ and $\xi \in H$, some $\delta > 0$ such that whenever $||\eta - \xi|| \leq \delta$ then $||\omega_\eta - \omega_\xi|| \leq \epsilon$. Let $\omega_{\xi, \eta}(x) = \langle x\xi, \eta\rangle$. Then if $||\eta - \xi|| \leq \delta$, we have $||\omega_\xi - \omega_{\xi, \eta}|| \leq ||\xi||\delta$. Similarly, $||\omega_\eta - \omega_{\xi, \eta}|| \leq ||\eta||\delta \leq (||\xi|| + \delta)\delta$. So $||\omega_\eta - \omega_\xi|| \leq (2||\xi|| + \delta)\delta$. Choose $\delta$ so that $(2||\xi|| + \delta)\delta \leq \epsilon$ and the claim is proved.