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In this recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C*-algebras. The answer, provided by Caleb Eckhardt in a comment, was that an amenable action on a unital C*-algebra leads to an amenable action on the compact spectrum of the center of said algebra, and hence exactness follows from a well known result.

The present question is pretty similar to my previous question, but I am now employing a different amenability condition, as follows:

Definition. An action $\theta$ of a discrete group $G$ on a C*-algebra $A$ is said to satisfy the approximation property if there exists a net $\{ a_i \}_{i\in I}$ of finitely supported functions $$ a_i: G \to A, $$ which is bounded in the sense that $$ \sup _{i\in I} \Big\Vert {\sum_{g\in G } a_i(g)^* a_i(g)} \Big\Vert < \infty , $$ and such that $$ \lim _{i \rightarrow \infty } \sum_{h\in G } a_i(gh)^* b\,\theta_ g( a_i(h)) = b,\quad \forall b\in A . $$

See the discussion before [1, Definition 20.11].

This condition is weaker than Anantharaman-Delaroche's usual condition of amenability (definition 4.3.1 in Brown and Ozawa) in that it does not require the $a_i$ to take values in the center of $A$. Nevertheless it is enough for most purposes and in particular it implies that the crossed product is nuclear provided $A$ is nuclear [1, Proposition 25.10].

Thus, here is the new version of my question:

Question: Suppose that a discrete group $G$ admits an action on a unital C*-algebra $A$, satisfying the above approximation property. Is $G$ necessarily exact?

Notice that, precisely because the $a_i$ are allowed to take values outside the center of $A$, it is not immediately clear that the above condition passes to the center, and hence Caleb's answer might no longer work in this case.

Reference:

[1] Partial Dynamical Systems Fell Bundles and Applications

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  • $\begingroup$ I should say that if $A$ is nuclear, then $G$ is indeed exact. This is because the reduced C*-algebra of $G$ will be a subalgebra of the crossed product, which is nuclear as mentioned above. $\endgroup$
    – Ruy
    Nov 21, 2016 at 22:51

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The answer to the question is positive: see Remark 6.6 in the paper

https://arxiv.org/pdf/1904.06771.pdf

The approximation property implies amenability in the sense of Claire Anantharaman Delaroche, this has been proved in

https://arxiv.org/pdf/1907.03803.pdf

for discrete groups, see also

https://arxiv.org/pdf/1904.06771.pdf

Indeed, it has been stablished recently that the approximation property is equivalent to amenability even in the more general context of locally compact groups by Ozawa-Suzuki in

https://arxiv.org/abs/2011.03420

and that the above question in this context also has a positive answer (see Corollary 3.6).

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