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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?

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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.

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  • $\begingroup$ Whoops sorry, yeah I should have clarified that what you wrote in (2) is often called nuclearity. Will update answer. So Connes gives (1) implies (2) and Haagerup gives (2) implies (1). $\endgroup$ – PStheman Oct 17 at 21:35
  • $\begingroup$ Sorry I was sleepy... By purity I meant nuclearity. Thank you again $\endgroup$ – Sui Oct 18 at 3:38
  • $\begingroup$ I thought nuclearity is when C* norm on $A\otimes B$ is unique for every $B$. $\endgroup$ – Sui Oct 18 at 4:09
  • $\begingroup$ You're right, this is yet another equivalence! $\endgroup$ – PStheman Oct 18 at 5:02
  • $\begingroup$ I didn't find the proof of this equivalence either... Is it easy to show? $\endgroup$ – Sui Oct 18 at 5:21

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