A countable discrete group $\Gamma$ is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions $(\phi\_n)$ on $\Gamma$ such that $\phi\_n\rightarrow 1$ pointwise and $\sup\_n ||\phi\_n||\_{cb}\leq 1$, where $||\phi||\_{cb}$ denotes the (completely bounded) norm of the Schur multiplier on $B(l\_2\Gamma)$ associated with $(x,y)\mapsto \phi(x^{-1}y)$.


A countable discrete group $\Gamma$ is exact if there exists a sequence of positive-definite kernels $\phi\_n:\Gamma\times \Gamma\rightarrow \mathbb{C}$ with the following two properties:


1 For every finite set $F\subseteq \Gamma$ and every $\epsilon>0$ there is an $N$ such that $g\_1^{-1}g\_2\in F\Longrightarrow |\phi\_n(g\_1,g\_2)-1|< \epsilon, \forall n>N$.


2 For every $n$ there is a finite set $F\subseteq \Gamma$ such that


$\phi\_n(g\_1,g\_2)\neq 0\Longrightarrow g\_1^{-1}g\_2\in F$. 


It is well-known that for discrete groups weakly amenability with Cowling-Haagerup constant 1 implies exactness. (More generally, AP implies exactness).


Is there a direct proof of this result without using the fact that the reduced group C\*-algebra $C^\*\_r(\Gamma)$ is exact as a C\*-algebra implies that $\Gamma$ is exact as a group?



Weakly amenability and exactness for discrete groups
See Theorem B/Theorem 3.4. in "Exactness of locally compact groups" <http://arxiv.org/abs/1603.01829>


