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If $I$ is a closed ideal in $C^*$ algebra $A$, then there is a unique $*$ homomorphism $\phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $I\to M(I)$, where $M(I)$ is the multiplier algebra of $I$.

My question is:If $A$ has a unit $e$, the map $\phi$ must not be zero since $e \mapsto (L_{e},R_{e})$.

If $A$ is not unital, can we conclude that $\phi$ is also a nonzero $*$ homomorphism?

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    $\begingroup$ Looking at the quantity of questions you post here, I suggest you to think more about them yourself and try to build knowledge. $\endgroup$ – user42761 Jul 25 '18 at 11:57
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If $\phi$ is zero, then also $I \to M(I)$

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