Equivalence between the GNS representation of two different positive linear functionals

Question

Let $\varphi $ be a positive linear functional on $C^*$-algebra $A$ and let $(\pi _{\varphi},H_\varphi ,\xi)$ be the associated GNS representation. Let $\psi \in A_+^*$. Show that the two next propositions are equivalent :

i) there exists $\eta \in H_\varphi$ such that $\psi (x)=\langle\eta,\pi _\varphi(x)\eta\rangle$ for all $x\in A$;

ii) there exists a sequence $(x_n)\subset A$ such that $\|\varphi _n -\psi\| \to 0$ where $\varphi _n(x)=\varphi (x_n^*xx_n)$.

I can show that i) $\Rightarrow$ ii) : since the GNS representation is cyclic there is a sequence $(x_n)$ such that $\pi _\varphi (x_n)\xi \to \eta $. Then $\varphi (x_n^*xx_n) =\langle\pi _\varphi(x_n)\xi,\pi _\varphi(x) \pi _\varphi(x_n)\xi\rangle$ and it is not too difficult to show that $\|\varphi _n -\psi \| \to 0$.

However the converse seems a little bit more difficult, there is no reason that $\pi _\varphi (x_n)\xi \to \eta $.

Answer 1

This issue is dealt with in the paper by R.V. Kadison, States and representations, Transactions of the Amer. Math. Soc. 103 (1962), 304--319. The main point is to show that the set of vector states of a representation of a $C^\ast$-algebra is a closed subset of $A^\ast$. The proof is quite involved. I would be glad to see a shorter argument if you have one.