Let $A$ and $B$ be C* algebras and let $\varphi : M(A) \to M(B)$ be a unital $*$-homomorphism of their multiplier algebras. Suppose, in addition, that $\varphi$ is strictly continuous (of course, norm continuity is automatic).
I'm looking for a proof of or counterexample to the following assertion:
If $\varphi$ is injective on $A$, then $\varphi$ is injective.
At first I thought this would be true and easy using that injective $\Leftrightarrow$ isometric for $*$-homomorphisms and that $A$ is strictly dense in $M(A)$. The problem with this approach is that the norm is generally discontinuous for the strict topology. Thanks.
Afterthought: as t.b.'s answer shows, most of the hypotheses here are quite besides the point. A $*$-homorphism $M(A) \to B$ is injective if and only if it is injective on $A$. More generally, a $*$-homomorphism $A \to B$ is injective if and only it is injective on some essential ideal $I$ of $A$.