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.


[1] Partial Dynamical Systems Fell Bundles and Applications

  • $\begingroup$ 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

1 Answer 1


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


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


for discrete groups, see also


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


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


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