Relative weak-star topology on pure states

Question

Let $A$ be a (unital) C*-algebra and consider $PS(A)$, the set of all pure states on $A$ with the relative weak-star topology. I would like to check (a weaker form of) Uryshon's lemma on $PS(A)$ in this sense: Let $E$ be a relative weak-star open set in $PS(A)$ and $\phi\in E$. Does there exists a positive element $a\leq1 $ in $A$ with $\phi(a)=1$ such that $\psi(a)=0$ for all pure states in the complement of $E$?

Answer 1

Let $A=M_2(\mathbb C)$. In this case, $$ PS(A)=\{\text{Tr}(P\cdot):\ P \text{ is a rank-one projection }\}. $$

Let $$ E=\{f\in PS(A):\ f(e_{11})<3/4\}. $$ Since $PS(A)$ is identified with the rank-one projections, $$ E=\{p\in M_2(\mathbb C):\ p^*p=p,\ \text{Tr}(p)=1,\ p_{11}<3/4\}. $$ Now let $\varphi$ the pure state corresponding to $p=\begin {bmatrix}1/2&1/2\\1/2&1/2\end {bmatrix} $, and $\psi $ the one corresponding to $p=e_{11} $. The $a $ you want has to satisfy $$0=\psi (a)=\text{Tr}(e_{11}a)=\text{Tr}(e_{11}ae_{11})=a_{11}. $$ As $a\geq0$, this implies that $a_{12}=a_{21}=0$, and so $a=\lambda e_{22}$, with $0\leq\lambda\leq1$. But then $$\varphi (a)=\lambda \,\varphi (e_{22})=\lambda \,\text {Tr}(pe_{22})=\lambda /2\leq1/2. $$