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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?

Sui
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1 Answers1

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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.

PStheman
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  • 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). – PStheman Oct 17 '20 at 21:35
  • Sorry I was sleepy... By purity I meant nuclearity. Thank you again – Sui Oct 18 '20 at 03:38
  • I thought nuclearity is when C* norm on $A\otimes B$ is unique for every $B$. – Sui Oct 18 '20 at 04:09
  • You're right, this is yet another equivalence! – PStheman Oct 18 '20 at 05:02
  • I didn't find the proof of this equivalence either... Is it easy to show? – Sui Oct 18 '20 at 05:21
  • I am reading On Nuclear C*-Algebras and I think I will find what I need here. – Sui Oct 18 '20 at 10:40
  • Its definitely not easy. But it can be found in the first couple of sections in Brown and Ozawa's book. – PStheman Oct 18 '20 at 13:48
  • Thank you a lot. I was reading An introduction to classification of amenable C*algebra by the way. The proposition was left in addenda without proof or reference... – Sui Oct 18 '20 at 14:14