approximately inner. Then for every finite set $F \subset {C^* (G, A, {\alpha})}$ and every ${\varepsilon}> 0$, there exists a unital surjective linear map $\psi \colon {C^* (G, A, {\alpha})}\to A$ such that:

1.  \[Prp\_magic\_a.1\] $\| \psi \| \leq {{\operatorname{card}}}(G)$.

2.  \[Prp\_magic\_a.2\] $\psi (a) = a$ for all $a \in A$.

3.  \[Prp\_magic\_a.3\] $\| \psi (xy z^*) - \psi (x) \psi (y) \psi (z)^*\| < {\varepsilon}$ for all $x, y, z \in F$.

Let ${\varepsilon}> 0$ and let $F \subset {C^* (G, A, {\alpha})}$ be a finite set. Set $$F_0= \Big\{ E_g (v), E_g (v)^* \colon g \in G \mbox{ and } v \in F \Big\},$$ $$M= \max \Big(1, \ \max_{a \in F_0}   \| a \|  \Big),
\qquad
\mbox{ and }
\qquad
{\varepsilon}'= \frac{{\varepsilon}}{2 M^2 {{\operatorname{card}}}(G)^3}.$$ Applying [Lemma \[Lem.s.a.i\_hom\]]{} with ${\varepsilon}'$ in place of ${\varepsilon}$ and $F_0$ as given, we get [homomorphism]{} $g \mapsto z_g$ from $G$ to $U (A)$ such that, for all $g \in G$ and $a \in F_0$, $$\label{Ap_inner.a}
\| {\alpha}_g (a) - z_g a z_g^* \| < {\varepsilon}'.$$ Now define $\psi \colon {C^* (G, A, {\alpha})}\to A$ by $$\sum_{g \in G} a_g u_g \mapsto \sum_{g \in G } a_g z_g .$$ Clearly $\psi$ is unital, linear, 