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Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps $\phi: A \rightarrow B$ and $L: B \rightarrow A$ such that $id ~{\approx}{\epsilon} ~L \circ \phi$ on $F$. Then is $A$ an amenable $C^{*}$-algebra?

I would be thankful if you help me out.

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    $\begingroup$ This follows immediately from the fact that a $C^\ast$-algebra is amenable if and only if it has the completely positive approximation property. $\endgroup$
    – Jamie Gabe
    Commented Nov 11, 2020 at 9:12
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    $\begingroup$ @JamieGabe Yes, you are completely right. I haven't CP app. property in my mind. Thanks for your help. $\endgroup$
    – Peg Leg Jonathan
    Commented Nov 11, 2020 at 9:30

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