Let $A$ be a $C^*$-algebra and $B,C$ two $C^*$-subalgebras of $A$. It is well-known that if $C$ is a closed ideal of $A$, then $B+C$ is a $C^*$-algebra of $A$.
This makes me think that in general $B+C$ need not be a $C^*$-subalgebra, but I couldn't find any counterexample. I guess we already have a problem because $B+C$ need not be an algebra. Are there any concrete examples?
In $M_2(\mathbb C)$, take $$ B=\biggl\{\begin{bmatrix} b&0\\0&0 \end{bmatrix}:\ b\in\mathbb C\biggr\},\qquad C=\biggl\{\begin{bmatrix}c&c\\ c&c \end{bmatrix}:\ c\in\mathbb C\biggr\}. $$ Then $B+C$ is not an algebra, since it doesn't contain $$\begin{bmatrix}1&1\\0&0\end{bmatrix}=\begin{bmatrix} 1&0\\0&0\end{bmatrix}\,\begin{bmatrix} 1&1\\1&1\end{bmatrix}.$$
For an abelian example, $A=\mathbb C^3$, and $$ B=\{(b,b,0):\ b\in\mathbb C\},\qquad C=\{(0,c,c):\ c\in\mathbb C\}. $$ Then $$ B+C=\{(b,b+c,c):\ b,c\in\mathbb C\} $$ is not an algebra, because $$ (0,1,0)=(1,1,0)(0,1,1)\in BC\setminus (B+C). $$