(The question below is inspired from the question Extending a $C^*$-algebras morphism to the multipliers and its answer.)
A Banach algebra A is said to be faithful when for each $a\in A$, $aA=(0) \Rightarrow a=0$ and for each $a\in A$, $Aa=(0) \Rightarrow a=0$.
Let $A$ be a Banach algebra (not necessarily a unital one). By a multiplier on $A$ is meant a pair $(L,R)$ of maps from $A$ to itself such that for each $a,b\in A$, $R(a)b=aL(b)$ (where juxtaposition stands for the product in the algebra $A$). When the algebra $A$ is faithful, it well-known that both $L,R$ are in fact linear and bounded. Moreover $M(A)$ is an algebra on its own, with identity $(\mathit{id}_A,\mathit{id}_A)$ and multiplication $(L',R')(L,R)=(L'\circ L,R\circ R')$. With the norm inherited from $\mathcal{B}(A,A)\times\mathcal{B}(A,A)$, $M(A)$ even is a Banach algebra. Finally $A$ embeds into $M(A)$ under $a\mapsto (L_a,R_a)$, where $L_a$ (resp. $R_a$) is the left (resp. right) translation by $a\in A$.
Now let $f\colon A\to M(B)$ be a bounded homomorphism of Banach algebras, where furthermore $B$ is assumed faithful and $f$ non-degenerate which means that the linear span of $f(a)^L(b),a\in A, b\in B$, is dense in $B$ and the linear span of $ f(a)^R(b),a\in A, b\in B$, is dense in $B$, with $f(a)=(f(a)^L,f(a)^R)\in M(B)$, $a\in A$.
Is it true that $f$ may be extended (uniquely) to a bounded homomorphism of algebras from $M(A)$ to $M(B)$? If not, what could be the missing assumptions (apart from $A,B$ being $C^*$-algebras)?
In fact it is not difficult to check that (using faithfulness) for each $(L,R)\in M(A)$ the correspondences $\sum_i f(a_i)^L(b_i)\mapsto \sum_{i}f(L(a_i))^L(b_i)$ from the linear span of $f(a)^L(b)$, $a\in A$, $b\in B$, to $B$, and $\sum_i f(a_i)^R(b_i)\mapsto \sum_i f(R(a_i))^R(b_i)$ from the linear span of $f(a)^L(b)$, $a\in A$, $b\in B$, to $B$, are both well-defined, and should correspond to the first factor and the second factor of $g(L,R)$, where $g$ is the would-be extension of $f$.
What I don't succeed in is to prove that these maps are bounded, and thus may be extended to the whole $M(A)$.