Lemma: Equal-snapshot separation

For $h=h^{\ast }\in M$ and a unital $C^{\ast }$-subalgebra $A\subseteq M$,

if and only if $\omega (h)=\nu (h)$ for every pair of states with $\omega {|}_A=\nu {|}_A$.

Proof. Membership gives equality by restriction. If $h\notin A$, real Hahn–Banach separation gives a bounded Hermitian functional $f$ vanishing on $A$, with $f(h)\ne 0$. Write $f=f_{+}-f_{-}$ using its Jordan decomposition. Because $1\in A$,

The states $f_{+}/m$ and $f_{-}/m$ have equal snapshots and different values on $h$, a contradiction. The Jordan decomposition used here is standard $C^{\ast }$-algebraic background Pedersen.