Theorem: Normal tensor-free comparison

Suppose the ambient algebras and local algebras are von Neumann algebras, their inclusions are normal, and $T$ is normal UCP. The three conditions of Theorem thm:comparison remain equivalent when snapshot determinacy is tested on all normal global input states. The unique induced local map $K$ is normal UCP. The factorization holds on preduals and is stable under normal independent matrix amplification.

Proof. Use Lemma lem:normalseparation in place of Lemma lem:separation. The matrix-positivity proof of complete positivity is unchanged.

To prove normality, let $b_{\lambda }$ be a bounded increasing net of positive local output elements, with supremum $b$. Normality of $T$ gives

in the ambient input algebra. These elements belong to the input von Neumann subalgebra, where the same supremum is computed. Thus $K(b_{\lambda })\uparrow K(b)$, proving normality. Dualization gives the predual identity.