We say a C$^{*}$-algebra $A$ is primitive if it has a faithful (injective) non-zero irreducible representation on some Hilbert space.
Is there a relatively easy (canonical) example of a C$^{*}$-algebra $A$ that is primitive, but whose quotient by some closed ideal is not primitive.
I am only now just starting to look into representations of C$^{*}$-algebras, so I am fairly new to this area. I know that for any Hilbert space $H$, $B(H)$ is primitive. I also know that every non-zero simple C$^{*}$-algebra is primitive. Thus, on a separable Hilbert space $H$, the Calkin algebra $B(H)/K(H)$ would still be primitive. I also know that an abelian C$^{*}$-algebra $A$ is primitive iff $A=\mathbb{C}$.
The Toeplitz algebra $\mathbf A$ acting on the Hardy space $H^2$ is primitive, it has $K(H^2)$ as an ideal, and $\mathbf A/K(H^2)\cong C(\mathbb T)$ (where $\mathbb T$ is the set of unit complex numbers) is not primitive.