Let $\varphi$ be a state of the $C^*$-algebra $A$, $B\subset A$ a hereditary subalgebra and $K_\varphi:=\{x\in B : 0\le x \le 1, \varphi(x)=1\}$. Let $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ be the GNS representation corresponding to $\varphi$. We can write $\varphi(x)=\langle\pi_\varphi(x)\xi_\varphi,\xi_\varphi\rangle$ for a cyclic vector $\xi_\varphi\in H_\varphi$ belonging to $\pi_\varphi$.
I already showed, that for every state $f\in S(A)$, $f\neq\varphi$, there exists $x\in K_\varphi$ such that $f(x)<1$. I now want to show, that for all $\eta\in H_\varphi$ with $\eta \perp \xi_\varphi$, there exists $x\in K_\varphi$ sucht that $\pi_\varphi(x)\eta=0$.
I have no idea how so solve this. I only know that $\langle\pi_\varphi(x)\eta,\zeta\rangle<1$ for any $\zeta\in H_\varphi$. I don't know what I'm missing.
Thank you in advance!
Edit: I found that $\varphi$ is a pure state. With $\varphi$ a pure state, this simply follows from Kadison's transitivity theorem.