step, $$\begin{aligned}
\| \alpha_g (a_j) - w_g a_j w^*_g\|
&\leq
\| \alpha_g (a_j) - \alpha_g (\varphi_{\infty, n} (b_j))\|
\\& \qquad +
\| \varphi_{\infty, n} (z_g) \varphi_{\infty, n} (b_j) \varphi_{\infty, n} (z^*_g)
-
w_g a_j w^*_g
 \|
\\\notag
&< \frac{{\varepsilon}}{2} + \frac{{\varepsilon}}{2} = {\varepsilon}.\end{aligned}$$ This completes the proof.

\[N\_9408\_StdNotation\_CP\] Let $A$ be a unital C\*-algebra and let ${\alpha}\colon G \to {{\operatorname{Aut}}}(A)$ be an action of a finite group $G$ on $A$. For $g \in G$, we let $u_g$ be the element of $C_{\mathrm{c}} (G, A, {\alpha})$ which takes the value $1$ at $g$ and $0$ at the other elements of $G$. We use the same notation for its image in $C^* (G, A, {\alpha})$. Also, for each $g \in G$, we define the map $ E_g \colon  {C^* (G, A, {\alpha})}\to A$ by $$E_g (a) = a_g,$$ where $a= \sum_{g \in G} a_g u_g$. We also denote by $A^{\alpha}$ the fixed point algebra, given by $$A^{\alpha} = \big\{ a \in A \colon
\alpha_g (a) = a \mbox{ for all } g \in G \big\}.$$

\[Prp\_magic\_a\] Let $A$ be a unital [C\*-algebra]{} and let $\alpha \colon  G \to {{\operatorname{Aut}}}(A)$ be an action of a finite group $G$ on $A$ which is strictly 