Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$.
Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$.
Let $H_0 = \{v \in H \ \vert \ a.v \in H \ \forall a \in \mathcal{A} \}$ and $\rho$ the left regular representation of $\mathcal{A}$ on $H_0$.
Question: Is $H_0$ a dense subspace of $H$ and $\rho$ faithful? Else what are the first counter-examples?
Also what would be the (minimal) additional assumptions for having $H_0$ dense and $\rho$ faithful?
Remark: the problem seems reduce to the type of closure on $\mathcal{A}$ (see the comments below).
Remark: for the Heisenberg algebra $\mathcal{A} = \langle a,b \ \vert \ [a,b]=1 \rangle$, then $H_0$ is dense and $\rho$ faithful, whereas $\mathcal{A}$ can't have a Banach structure (see here).