Is a crossed product of a separable $C^\ast$-algebra by a finite group separable?

Question

If $A$ is a separable $C^\ast$-algebra, $\alpha$ is an action on $A$ by a finite group $G$, then is the crossed product $A\rtimes_\alpha G$ separable?

Answer 1

Of course. Because $C_r(\mathcal A,G)$ is the C$^*$-algebra generated--in the right environment--by $\mathcal A$ and $G$. You take a dense subset $\mathcal A_0$ of $\mathcal A$, and then all sums $$\sum_{j=1}^m a_jg_j,$$ are dense, where $a_j\in\mathcal A_0$ and $g_j\in G$.

As you can see, you don't need $G$ finite: countable suffices.