It is known that a discrete group $G$ is exact if and only if it admits an amenable action on a compact topological space. Would it also be true that $G$ is necessarily exact when it admits an amenable action on a unital C*-algebra?
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1$\begingroup$ Could your remind us of the definition of what it means for a G-action on a noncommutative Cstar-algebra to be amenable? Is this the definition in e.g. Brown-Ozawa's book? $\endgroup$– Yemon ChoiCommented Sep 28, 2016 at 14:12
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$\begingroup$ @Yemon, you are 100% correct. This was originally formulated by Anantharaman-Delaroche following ideas of Zimmer. $\endgroup$– RuyCommented Sep 28, 2016 at 16:33
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2$\begingroup$ Doesn't it follow from the definition (4.3.1 in Brown and Ozawa) that if G acts amenably on a unital C*-algebra A then it also acts amenably on the center of A, which provide an amenable action on a compact space? $\endgroup$– Caleb EckhardtCommented Sep 28, 2016 at 20:58
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$\begingroup$ Yep, you are right @Caleb. This settles it! $\endgroup$– RuyCommented Sep 28, 2016 at 21:39
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$\begingroup$ @Caleb, you will probably see this, but in any case let me point out that I am posting a new version of this question in mathoverflow.net/questions/255246/… $\endgroup$– RuyCommented Nov 21, 2016 at 16:09
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