A question on von Neumann algebra

Question

Let $M$ be a von Neumann algebra and let $\pi: M \rightarrow B(H)$ be a representation. Suppose that $N$ is a von Neumann subalgebra of $M$ such that $E$ is a faithful conditional expectation from $M$ to $N$. Is it true that if $\pi$ is faithful on $N$ implies that $\pi$ is faithful on $M$?

Answer 1

No.

Let $M=M_2(\mathbb C)\oplus\mathbb C$, and $N=\mathbb C\,I_3$.

Let $\pi:M\to M_2(\mathbb C)$ given by $\pi(A\oplus b)=b\,I_2$. Take $E(A,b)=\tfrac23\,\operatorname{Tr}(A)+\tfrac13\,\operatorname{Tr}(b)$.

Then $E$ is a faithful conditional expectation onto $N$, $\pi$ is faithful on $N$, and $M_2(\mathbb C)\oplus0\subset\ker\pi$.