Corollary: Exact algebraic interface theorem

For effects restricted to $\mathcal{A}$,

All such effects agree if and only if $E_{\mathcal{A}}(X_J)=0$. The corresponding Helstrom projector belongs to $\mathcal{A}$.

Proof. For $M\in \mathcal{A}$, self-adjointness gives $\mathrm{tr}(M{X}_J)=\mathrm{tr}(M{E}_{\mathcal{A}}(X_J))$. The spectral projector is admissible by Lemma lem:expectation. Apply Lemma lem:helstrom.