Lemma: Separation by equal-snapshot states

Let $\mathfrak{A}\subseteq \mathfrak{M}$ be a unital $C^{\ast }$-subalgebra. For $h=h^{\ast }\in \mathfrak{M}$, the following are equivalent:

1. $h\in \mathfrak{A}$;

2. $\omega (h)=\nu (h)$ whenever $\omega ,\nu \in \mathsf{S}(\mathfrak{M})$ have equal restrictions to $\mathfrak{A}$.

Proof. The forward implication follows by restriction.

Suppose $h\notin \mathfrak{A}$. The real closed linear subspace ${\mathfrak{A}}_{\mathrm{sa}}$ does not contain $h$. Real Hahn–Banach separation gives a bounded real functional $\ell$ on ${\mathfrak{M}}_{\mathrm{sa}}$ such that

Its complexification is a bounded Hermitian functional $f$ on $\mathfrak{M}$, vanishing on $\mathfrak{A}$.

Write $f=f_{+}-f_{-}$. Because $1\in \mathfrak{A}$, we have

The nonzero value $f(h)$ implies $m>0$. Moreover,

Thus

are states with equal snapshots on $\mathfrak{A}$, but

This contradicts (ii).