Consider the following question:
Let $A$ be a normed space containing a closed subset $B\subseteq A$ and a dense subset $D\subseteq A$. Is $B \cap D$ necessarily a dense subset of $B$?
My conclusion is that it need not hold in general. To see this, take $A=\mathbb{R}$ (with the standard metric), $B=\{\pi\}$ and $D=\mathbb{Q}$. Then $B\cap D = \{\pi\} \cap \mathbb{Q} = \emptyset$, the closure of which is $\emptyset$. Therefore $B\cap D$ is not dense in $B$.
I am particularly interested in understanding what happens if we add the assumption that $A$ is a (unital) C*-algebra, $B$ is a sub-C*-algebra of $A$ (hence closed) and $D$ is a dense $*$-subalgebra of $A$. Is the answer affirmative then? Can anyone come up with a counterexample?