I wish to show the following: Let $\Gamma$ be a discrete group, $H$ a subgroup of $\Gamma$ and $\mathcal{A}$ be a unital $\Gamma$-$C^*$-algebra. I want to prove that the map $E_H: C_c(\Gamma,\mathcal{A})\to C_c(\Gamma,\mathcal{A})$, defined by $$E\left(\sum_s a_ss\right)=\sum_{s\in H}a_ss$$ extends to a continuous map from $\mathcal{A}\rtimes_r\Gamma$ onto $\mathcal{A}\rtimes_r H$.
I follow the argument illustrated in Brown-Ozawa which shows that the canonical conditional expectation $E:\mathcal{A}\rtimes_r\Gamma\to \mathcal{A}$ is continuous
Without any loss of generality, let us say that $\mathcal{A}$ has a faithful covariant representation inside $\mathbb{B}(\mathcal{H})$. I resort to using Fell’s absorption principle. Using this principle, we view $\mathcal{A}\rtimes_{\alpha,r}\Gamma$ as a sub-algebras of $\mathbb{B}(\mathcal{H})\otimes C_r^*(\Gamma)$.
Now, under this identification, that $$E_H=(\text{id}\otimes\mathbb{E}_H)|_{\mathcal{A}\rtimes_r\Gamma}$$
,where $\mathbb{E}_H:C_r^{\star}(\Gamma)\to C_r^*(H)$ is the canonical conditional expectation.
Please let me know if I missed something here.
Thank you for the help.