Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0\to A\to C\to B\to 0$$
Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?
Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?
Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?