Is the space of positive linear functionals on a $C^{*}$-algebra $\mathcal{A}$ closed in the norm topology of $\mathcal{A}^{*}$?

Question

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\mathcal{A}^{*}$ its Banach dual. Let $\mathcal{P}\subset \mathcal{A}^{*}$ be the convex set of positive linear functionals on $\mathcal{A}$. Is $\mathcal{P}$ closed in the norm topology?

Answer 1

The answer is positive if your $C^*$ algebra $A$ has a unit, say, $e$. In that case, there is a theorem asserting that a bounded functional is positive on $A$ if and only if its norm is attained at $e$. Therefore, if $f_n\to f$ in norm, then $|f_n(e)-f(e)|\to 0$, and also $f_n(e)=\|f_n\|\to\|f\|$, hence $f(e)=\|f\|$ so $f$ is a positive functional.