Restriction of a faithful representation of a von Neumann algebra

Question

Suppose that $M$ is a von Neumann algebra represented on a Hilbert space $H$. Suppose $H_0$ is a subspace of $H$ invariant under $M$ so that we get a representation $\pi: M \rightarrow B(H_0) $. Will $\pi$ be faithful? I do not much about von Neumann algebras. Is representation of $M$ on $B(H)$ is faithful?

Answer 1

No. Take $H=M=\mathbb C\oplus\mathbb C$, $H_0=\mathbb C\oplus0$.

Not sure what you mean by the last question.