Lemma: Normal equal-snapshot separation

Let $\mathfrak{A}\subseteq \mathfrak{M}$ be a unital von Neumann subalgebra. If $h=h^{\ast }\notin \mathfrak{A}$, there exist normal states $\omega ,\nu$ on $\mathfrak{M}$ with equal restrictions to $\mathfrak{A}$ and $\omega (h)\ne \nu (h)$.

Proof. The self-adjoint part of $\mathfrak{A}$ is ultraweakly closed. Separation in the dual pairing $(\mathfrak{M},{\mathfrak{M}}_{\ast })$ gives a nonzero normal Hermitian functional $f$ vanishing on $\mathfrak{A}$ with $f(h)\ne 0$. Its Jordan components $f_{\pm }$ are normal. As in Lemma lem:separation,

Normalize them to obtain the stated normal states.