Let $A$ be a C*-algebra.
1, $A$ is said to be amenable if every derivation from $A$ to some dual Banach $A$-bimodule is inner.
2, $A$ is said to be amenable if for every finite set $F\subset A$ and $\epsilon>0$ there is some $M_n$ and contractive completely positive linear maps $\phi:A\to M_n$ and $L:M_n\to A$ such that $\|(L\circ \phi)(a)-a\|<\epsilon $ holds for every $a\in F$
Does someone know where I can find the proof?
Note that (2) is often referred to as nuclearity.
Connes proved that all amenable C${}^*$-algebras are nuclear (atleast in the separable setting) and then Haagerup proved the converse. The papers are "On the cohomology of operator algebras" (1 implies 2) and "All nuclear C${}^*$-algebras are amenable" (2 implies 1) respectively. Both these papers make use of the theory of von Neumann algebras (in particular the fact that nuclear C*-algebras have injective double duals).
I'm not sure if there is a more modern proof but the mentioned papers are quite short, so it might be worth taking a look there.