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.



A perturbation question for the intersection of C*-subalgebras
If $q(A)$ is closed in $M/B$, then its preimage $A+B$ is closed in $M$. Here is an example when it is not: Let $M=C([0,1],M\_2)$, let $p,q\in M$ be the rank one projections 
$$
p=
\begin{pmatrix}
1 & 0\\
0&0
\end{pmatrix},
q(t)=
\begin{pmatrix}
1-t & t^{1/2}(1-t)^{1/2}\\
t^{1/2}(1-t)^{1/2} & t
\end{pmatrix}.
$$
Let $A$ be the elements of the form $fp$, with $f\in C[0,1]$ and $B$ the element of the form $gq$, with $g\in C[0,1]$ (i.e., $A=pMp$, $B=qMq$). Then the elements of $A+B$ have the form 
$$
fp+gq=
\begin{pmatrix}
f+g\cdot (1-t) & gt^{1/2}(1-t)^{1/2}\\
gt^{1/2}(1-t)^{1/2} & gt
\end{pmatrix}.
$$
Observe that the rate of decay near zero of the bottom right corner is at least linear. So choosing functions $g\_n\in C[0,1]$ such that $g\_nt^{1/2}\to t^{1/4}$ and $f\_n=-g\_n\cdot (1-t)$ we get the matrix 
$$
\begin{pmatrix}
0 & t^{1/4}(1-t)^{1/2}\\
t^{1/4}(1-t)^{1/2} & t^{3/4},
\end{pmatrix}
$$
in the closure but not in the algebraic sum.
 The subalgebras $A$ and $B$ don't share $M$'s unit but this can be fixed by adding the unit to them: the elements of $A+B+\lambda 1\_2$ have $\lambda+gt$ in the bottom right corner, so the same matrix as before does not belong to it.


