Let $A$ be a $C^{*}$-algebra. In what follows, a representation $\pi$ on $A$ is a $*$-homomorphism $\pi: A \to \mathscr{B}(\mathscr{H})$, the Banach space of bounded linear operators on a complex Hilbert space $\mathscr{H}$.
From the definition of a representation, it follows that every representation is continuous and $\|\pi(a)\|\le \|a\|$. Hence, $\pi(A)$ is a closed subspace of $\mathscr{B}(\mathscr{H})$, from where it follows that it is a $C^{*}$-subalgebra of $\mathscr{B}(\mathscr{H})$.
It is also easy to prove that $\operatorname{ker}\pi = \pi^{-1}(\{0\})$ is an ideal of $A$. Thus, one can consider the quotient algebra $A/\operatorname{ker}\pi$. Let $p$ be the canonical projection $p: A \to A/\operatorname{ker}\pi$.
In Bratteli & Robinson's book, it is stated that every representation $\pi$ on $A$ induces a new faithful representation on $A/\operatorname{ker}\pi$.
From the above analysis, I suppose this faithful representation is to be defined by $\tilde{\pi}: A/\operatorname{ker}\pi \to \mathscr{B}(\mathscr{H})$, with $\tilde{\pi} = \pi \circ p^{-1}$.
Is the analysis correct?