Representation of the multiplier algebra of a crossed product If $G$ is a locally compact group and it acts on a C*-algebra $A$, then there is a canonical *-homomorphism (injection) from the reduced crossed product $A\rtimes G$ to the adjointable operators on the Hilbert $A$-module $A \otimes L^2(G),$ $B(A \otimes L^2(G)).$ Is there a canonical *-homomorphism from the multiplier algebra $M(A\rtimes G)$ to $B(A \otimes L^2(G))$?
 A: The answer is yes, and it is indeed a special case of a much more general result:
If $B$ and $C$ are  C*-algebras, $E$ is a right Hilbert module over $C$,    and $\pi :B\to L(E)$ is a non-degenerate
*-homomorphism, then $\pi$ extends uniquely to a *-homomorphism $\tilde\pi $ from the multiplier algebra $M(B)$ to $L(E)$.
The argument is based on the fact that the non-degeneracy of $\pi $ guarantees that the subset of $E$ formed by the
elements of the form
$$
  x=\sum_{i=1}^n \pi (b_i)x_i,
  $$
with $b_i\in  B$, and $x_i\in  E$,
is dense in $E$.  Therefore, given $m$ in $M(B)$, one may define
$$
  \tilde\pi (m)x =  \sum_{i=1}^n \pi (mb_i)x_i.
  $$
In case $\pi $ fails to be non-degenerate, the uniqueness of $\pi $  may no longer be guaranteed, but the
existence results persists.   To see why, one may restrict each $\pi (b)$ to the essential space
$$
  E_{\text{ess}}:= \overline{\text{span}}\{\pi (b)x: b\in  B, \ x\in  E\},
  $$
and then the representation becomes non-degenerate and the above applies,
