Representation of the multiplier algebra of a crossed product

Question

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 $B(A \otimes L^2(G))$?

Answer 1

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,