Inclusion of multiplier algebras

Question

If $B$ is a $C^*$-subalgebra of $A$, i.e there exists an inclusion map $\phi\colon B \rightarrow A$, can we conclude that there exists a $*$-homomorphism beween the multiplier algebras $\bar{\phi} \colon M(B)\rightarrow M(A)$ ?

Answer 1

Yes, if $B$ sits inside $A$ non-degenerately. See [Prop. 2.1, Hilbert C*-Modules: A Toolkit for Operator Algebraists].

More precisely: If $(e_i)$ is an approximate unit for $B$ one may define $$ \bar \phi(x)\big (\phi(e_i)a \big) = \phi(xe_i)a \qquad ( x \in M(B), a \in A). $$

Add: If $\phi \colon A \to B$ is any surjective $*$-hom. you also have an extension to the multiplier algebras.