Is a unital $*$-homomorphism preserving a state is one-to-one?

Question

Let $M$ be a von Neumann algebra and let $\varphi$ be a faithful normal state on $M$. Suppose that $T \colon M \to M$ is a normal unital $*$-homomorphism preserving $\varphi$, i.e. $\varphi \circ T =\varphi$.

Is $T$ is one-to-one (injective)?

Answer 1

This appears to hold for a lot weaker conditions.

Let $A$ be a $C^*$-Algebra and $T: A \to A$ a * morphism that preserves a faithful state $\phi$ in the sense that $\phi \circ T = \phi$.

A state $\phi: A \to \mathbb{C}$ is faithful if for positive $X \in A$ we have $\phi(X)=0 \iff X=0$.

The kernel $K$ of $T$ is again a $C^*$-Algebra since * morphisms are continuous. Let $X \in K$, then $X^*X$ is positive. $\phi(X^*X)=\phi(T(X^*X))=\phi(0)=0$. Since $\phi$ is faithful this implies that $X^*X=0$ and we get $K=0$, i.e. the map $T$ must be injective.