Let $\mathcal A$ be a unital C* algebra.
Which properties does $\mathcal B \subset \mathcal A$ has to have for it to make sense to form the quotient algebra $\mathcal A / \mathcal B$?
In cases where this construction makes sense, does $\mathcal A / \mathcal B$ have any special structure/properties that are helpful? Put differently, for what kind of standard questions does it help to consider $\mathcal A / \mathcal B$ because it has desired properties?
If $\mathcal{B}$ is a closed $*$-ideal of $\mathcal{A}$, then $\mathcal{A}/\mathcal{B}$ is again a unital C*-algebra. This can be found in every introduction to C*-algebras. And of course this basic construction is used everywhere, so that it doesn't make sense to start a list of applications here. A well-known example is the Calkin algebra. In the commutative case, we have $C(X)/I(A) \cong C(A)$ for closed subspaces $A \subseteq X$ with vanishing ideal $I(A)=\{f : f|_A=0\}$.