Let $I$ be a proper closed ideal of a $C^*$-algebra $A$ and let $B$ be a unital (for simplicity) $C^*$-algebra. We have a natural inclusion $$I \otimes B \subseteq A \otimes B$$ Here, the tensor products are the spatial ones (minimal tensor product).
Can it ever happen that $I \otimes B = A \otimes B?$
I tried the following:
Let $a \in A \setminus I$. We want to show that $a \otimes 1_B \notin I \otimes B$. It is easy to see that $a \otimes 1_B \notin I \odot B$ (algebraic tensor product), but how can we ensure $a \otimes 1_B$ is not a limit of elements in $I \odot B$?
Considering the quotient map $q: A\to A/I$, one has that $$ q\otimes \text{id} : A\otimes B \to (A/I)\otimes B $$ is a well defined, nonzero map, vanishing on $I\otimes B$. Therefore $I\otimes B\subsetneq A\otimes B.$