Is every action from an amenable group amenable on a unital $C^*$-algebra?

Is every action from an amenable group amenable on a unital $C^*$-algebra?

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It would help if you reminded people what you mean by an amenable action on a $C^*$-algebra, and made more precise the quantifiers in your question. – Yemon Choi Oct 16 '10 at 4:33
Ah, so you were asking if each (discrete?) amenable group has an amenable action on some unital C*-algebra. This seems to be answered below. – Yemon Choi Oct 16 '10 at 7:23
Yes I mente discrete group. Sorry – m07kl Oct 16 '10 at 8:40

Yes it is. It follows from Theorem 3.3 of [1] and the fact that the trivial action of an amenable group on $\mathbb{C}$ is amenable. More modern reference is [2] (in particular Section 4.3).

[1] C. Anantharaman-Delaroche. Systèmes dynamiques non commutatifs et moyennabilité. Math. Ann., 279(2):297–315, 1987.

[2] N. P. Brown and N. Ozawa. C*-algebras and finite-dimensional approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008.

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Thanks for reference – m07kl Oct 16 '10 at 8:35
I think this is more easier to see if we use definition in [2] Section 4.3. – m07kl Oct 16 '10 at 8:40