Let $A$ be a unital $\mathrm{C}^*$-algebra. There are some well-known facts about the state space $S(A)$ and the set of pure states $PS(A)$. We might remark that in general $PS(A)$ is not weak$*$-closed but if we take the weak star closure, the pure state space, $\overline{PS(A)}^{w*}$, we have have a weak$*$-compact space.
If $f\in A$ is embedded in the second dual $\imath:A\hookrightarrow A^{**}$ via the Gelfand transform (normally only done for commutative $\mathrm{C}^*$-algebras),
$$\widehat{f}(\rho):=\rho(f)\qquad (\rho\in A^*),$$
then the nature of weak*-convergence of a net of states $\rho_\lambda\rightarrow \rho\in A^*$ implies that $\hat{f}$ is a continuous function from $S(A)\rightarrow \mathbb{C}$. The multiplication $\widehat{f}\otimes \widehat{g}\mapsto \widehat{fg}$ in $\imath(A)\subset C(S(A))$ is not the pointwise multiplication.
As an example consider $A=M_2(\mathbb{C})$, whose state space is parameterised by the closed unit ball in $\mathbb{R}^3$. It turns out that $\imath(A)\subset C(S(A))$ consists of all the affine functions, $(x,y,z)\mapsto ax+by+cz+d$.
Question 1: In general, is there any hope of generally describing which continuous functions arise in $\imath(A)\subset C(S(A))$?
If we restrict $\widehat{f}$ to $\overline{PS(A)}^{w*}$ we have a continuous function $\tilde{f}\in C(\overline{PS(A)}^{w*})$ on the pure state space.
If we restrict $\tilde{f}$ to $f_0\in C(PS(A))$, then we can extend $f_0$ by linearity to the (not closed) convex hull $\operatorname{co}PS(A)$.
Question 2: Is every continuous function $\widehat{f}\in\imath(A)$ determined by its restriction to $PS(A)$ (or perhaps $\overline{PS(A)}^{w*}$), in the sense that it is determined uniquely by linear/convex extension?
In the finite commutative case, the pure states $PS(A)$ are a finite set, every function $F\in C(S(A))$ is of the form $\widehat{f}$ for some $f\in A$, and all of these functions are got by linearly extending $F_{\left|PS(A)\right.}$ to $S(A)$.
I understand there are studies that try and recover a $\mathrm{C}^*$-algebra from the state and structures on it. I am not trying to be so comprehensive: just trying to describe a way to consider elements of a noncommutative $\mathrm{C}^*$-algebra as continuous functions on a compact Hausdorff space.
Thanks to Ruy I was able to find some references such as Landsman (p.18-19) which discusses an answer to Q. 1 and Q. 2 is yes.