I have a confusion regarding the unit cyclic vectors in the GNS construction of pure states. It seems every corresponding GNS representation $(\mathcal{H}_f, \varphi_f)$ of a pure state $f$ of a unital C*-algebra $\mathcal{A}$, should have its unique unit cyclic vector $x_f \in \mathcal{H}_f$. However, I also see a claim (forgot where) that for every non-zero unit vector $x\in \mathcal{H}_f\setminus \{0\}$, such that $\|x\|=1$, we always have, for $a\in\mathcal{A}$,
$$f(a)=\left\langle \varphi_f(a)x,x\right\rangle.$$
So for example, if $f(a)>0$ for every pure state $f$, we can conclude that $\varphi_f(a)$ is always a self-adjoint positive definite operator. Then, deduce that $a$ is positive in $\mathcal{A}$.
But is this correct? Shouldn't this equality only hold for the unique $x_f$ obtained from the GNS construction? Am I missing something? Is this a special property for pure states?
Best Answer
You don't say where, so I cannot comment on that.
For any $b\in A$ such that $f(b^*b)=1$ you have that $\hat b=\phi_f(b)x_f$ is a unit vector, and $$ \langle \phi_f(a)\hat b,\hat b\rangle=\langle\phi_f(a)\phi_f(b)x_f,\phi_f(b)x_f\rangle=\langle\phi_f(b^*ab)x_f,x_f\rangle=f(b^*ab). $$ So the assertion is not true as stated.
Now, if $f(a)≥0$ for all pure states then $a$ is positive. This follows from the fact that the state space is the weak$^*$ closure of the convex hull of the pure states. Then $f(a)≥0$ for all states and so $a≥0$.