In [this](https://mathoverflow.net/questions/250926/characterization-of-exact-groups-via-the-existence-of-amenable-actions-on-unital) 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](https://arxiv.org/abs/1511.04565)



Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2
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).


