Proposition: Expectation-based certificate

If such an $E$ exists, then the conditions of Theorem thm:comparison are equivalent to

In that case $K=ET{|}_{{\mathfrak{A}}_{\mathrm{out}}(Y)}$. Also, every local state $\eta$ has the common affine extension $\eta \circ E$.

Proof. The fixed-point range of $E$ is exactly $\mathfrak{A}$. Thus the displayed equality is equivalent to containment. Composition with $E$ then gives the unique local channel. Since $E{|}_{\mathfrak{A}}=\mathrm{id}$, $\eta \circ E$ restricts to $\eta$.