Injective von Neumann algebra

Question

Let $M$ and $N$ be von Neumann algebras equipped with faithful normal states. Let $\pi \colon M \to N$ be a normal unital injective $*$-homomorphism preserving the states and commuting with the modular groups. I know that in this situation there exists a faithful normal conditional expectation from $N$ onto $\pi(M)$ preserving the states.

Suppose that $N$ is injective. Is $M$ injective ?

Answer 1

Yes. If $A\subset B$ and $\phi:A\to M$ is completely positive, then $\pi\circ\phi:A\to N$ is completely positive. By the injectivity of $N$, there exists $\psi:B\to N$ such that $\psi|_A=\pi\circ\phi$. By the remaining conditions you mention, there is a conditional expectation $E:N\to \pi(M)$. Then $\pi^{-1}\circ E\circ\pi\circ\phi:B\to M$ is completely positive, and for any $a\in A$ we have $$ \pi^{-1}(E(\pi(\phi(a))))=\phi(a). $$

Answer 2

Alternatively to Martin's answer: $M \subset B(H)$ is in the range of a cp projection $P:B(H) \rightarrow M$ iff $M$ is injective. Composing the projection $P:B(H) \rightarrow N$ with $E: N \to M$ gives a cp projection for $M$.