In the proof for the theorem in the picture snipped off the book State Spaces of Operator Algebras by Erik M. Alfsen & Frederic W. Shultz, where is the Hahn-Banach Theorem is applied? As far as I can see, only Stone-Weierstrass Theorem is applied, which follows $\{\hat{a}|a\in A\}$ is dense in $C(K)$:
- $\hat{1}(\rho)=\rho(1)=1, \forall \rho\in K$, so, $\hat{1}$ is a constant function in $C(K)$;
- For distinct states $\mu,\nu$ on $A$, of course we can find some $a\in A$ such that $\mu(a)\neq \nu(a)$, so, $\hat{a}(a\in A)$ separate the points of $K$.
Since $\hat{a}(a\in A)$ is affine on $K$, $C(K)=A(K)$.
