Restriction of an irreducible representation

Question

Let $A$ be a C*-algebra and $\pi:A \to B(H)$ be a irreducible representation. Could we claim $\pi_{|B}$ is an irreducible representation if $B$ is a C*-subalgebra of $A$ ?

Answer 1

Definetely not. Just take $B$ to be the kernel of $\pi$.

For instance, $A=\mathbb C\oplus\mathbb C$, $B=0\oplus \mathbb C$, $\pi(a,b)=a$.

Even if you require $B$ to be unital with the same unit as $A$, the answer is no: let $$ A=M_2(\mathbb C),\ \ \ B=\left\{\begin{bmatrix}a&0\\0&b\end{bmatrix}:\ a,b\in\mathbb C\right\} $$ with $\pi$ the identity representation.