Proposition 43.4.8.label Let $M, N$ be von Neumann algebras, $\omega \in M^{*}$ and $\psi \in N^{*}$ be normal positive linear functionals with $\psi$ being faithful, and $\phi: M \to N$ be a positive linear map with $\psi \circ \phi \le \omega$, then $\phi$ is normal.
Proof, [Proposition 2.5.11, AP]. Let $\varphi = \psi \circ \phi \le \omega$, then $\varphi$ is normal by Lemma 43.4.5. For any bounded, upward-directed family $A \subset M$,
\[\psi(\sup(\phi(A))) =\sup(\psi(\phi(A))) = \sup(\varphi(A)) = \varphi(\sup(A)) = \psi(\phi(\sup(A)))\]
As $\psi$ is faithful, $\sup(\phi(A)) = \phi(\sup(A))$. Therefore $\phi$ is normal by (1) of Definition 43.4.6.$\square$
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