Let $A$ be a $C^*$-Algebra and $A''$ its enveloping von Neumann Algebra. Is the state space $S(A)$ of $A$ weak*-dense in the state space of $S(A'')$? I Know that every state on $A$ extends as a vector-state to a normal state on $A''$ and that $A$ is weak*-dense in $A^{**}\cong A''$. But I don't seem to get how you can approximate a state in $S(A'')$ by an state in $S(A)$.
Thank you in advance!
Your problem can reformulated as asking whether the positive part of the unit ball of $A^{\ast \ast}$ can be approximated (in the $\sigma(A^{\ast \ast \ast}, A^{\ast})$-topology) by elements in the positive part of $A^{\ast \ast}$.
If you remove the positivity assumption this is usually called Goldstine theorem and holds true for every Banach space, not just duals of $C^\ast$-algebras. You can try to extend the proof to positive functional, it uses a Hahn-Banach separation argument, see this notes[p97] therefore it should work.