# Why are these two definitions of amenability equivalent?

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?

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 thought nuclearity is when C* norm on $A\otimes B$ is unique for every $B$. – Sui Oct 18 at 4:09