Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra.
If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for $B$ (it is clear).
I want to show that $K: M(A) \to M(B)$ such that $K((L,R))=(\phi(L),\phi(R))$ is a surjective $∗$-homomorphism.
I can prove that $K$ is $∗$-homomorphism and I do not use separable condition.
How can I prove that $K$ is surjective?
Should I use separable condition?