Let $A$, $B$ be C$^*$-algebras, $\mu$ be a state on $B$ and $\mathcal{I}$ be a family of ideals in $A$. Let $I_0:=\cap_{I\in\mathcal{I}} I$ and put $A_0:=A/I_0$. Consider the minimal tensor product on $B\otimes A_0$. Furthermore, let $H_\mu$ be the GNS space of $B$ related to $\mu$ and denote by $H_\mu\otimes A_0$ the completion of $H_\mu\otimes_{alg} A_0$ under the norm coming from the $A_0$-valued inner product given by $\langle \xi\otimes a,\eta\otimes b \rangle=\langle \xi,\eta \rangle a^*b$. (In this way $H_\mu\otimes A_0$ becomes a Hilbert C$^*$-module over $A_0$.)
We can see $B\otimes A_0\subset H_\mu\otimes A_0$. I would like to characterize this subset. More specifically, I wonder if the following is true. For $x\in H_\mu\otimes A_0$: $$ x\in B\otimes A_0 \quad\Leftrightarrow\quad (id\otimes p_I)(x)\in B\otimes A/I \text{ for all } I\in\mathcal{I}, $$ where we denoted by $p_I:A_0\rightarrow A/I$ the canonical projection. Left to right is of course obvious, but I have no idea about the other direction...
