This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras".
Let M be a unital C*-algebra and let A, B be unital, closed star-subalgebras (so they are C
*-algebras in their own right). Let q be the quotient map of Banach spaces from M onto M / B. Is q(A) necessarily closed in M / B ?
(Of course if B were an ideal then the answer is yes, because any star-homomorphism between C*-algebras has closed range.)
If it makes any difference, I'm primarily interested in the case where M is a von Neumann algebra and A and B are sub- von Neumann algebras.