the induced trivial $*$ homomorphism

Question

Let $A=c_{0}\oplus \mathbb{K}$,$I=c_{0}$ is the closed ideal of $A$,there is an induced $*$ homomorphism $\phi:A/I\rightarrow M(I)/I$,where $M(I)$ is the multiplier algebra of $I$.$\phi(a+I)=(L_{a},R_{a})+I$.I have no idea about showing $\phi=0$.

Answer 1

It follows by the universal property of the multiplier algebra.

More generally, let $A,I,B$ be C*-algebras and $A = I \oplus B$. Then $I$ is an ideal in $A$ and the inclusion $I \to M(I)$ can be uniquely extended to a $*$-hom. $A \to M(I)$. However, such an extension is given by the projection $\pi_I : I \oplus B\to I$. So the map $$ A \to M(I) /I $$ is already $0$ and therefore also the map $A/I \to M(I)/ I$.