I am working on a problem on page 120 of the book C*-algebras and Finite-dimensional Approximations. The problem asks me to prove that if the action from the discrete group $\Gamma$ to the algebra $A$ i.e. $\tau\ \colon \Gamma\ \to Aut(A)$ is trivial, then $A\rtimes_{\tau,r}C^{\ast}_r(\Gamma)\cong A\otimes C^{\ast}_r(\Gamma)$. I can prove this by using the Fell's absorption principle. My problem is the full crossed version, say: if the action is trivial, whether $A\rtimes_{\tau}C^{\ast}(\Gamma)\cong A\otimes C^{\ast}(\Gamma)$? I guess so. But I do not know how to prove that.
Thanks for all help!