Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-inner product $ [\cdot,\cdot] $ on $ B $ by $$ \forall \xi,\eta \in \mathcal{E}: \quad [\xi,\eta] \stackrel{\text{df}}{=} \phi(\langle \xi,\eta \rangle_{B}), $$ where $ \langle \cdot,\cdot \rangle_{B} $ denotes the $ B $-valued inner product on $ \mathcal{E} $.

Question:Is it necessarily true that $ \mathcal{E} $ is already complete with respect to the metric induced by $ [\cdot,\cdot] $? If the answer is ‘no’, then is there a judicious choice of a faithful state $ \phi $ on $ B $ that would make this true?

Thank you very much.