# Weakly amenability and exactness for discrete groups

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?

## 1 Answer

See Theorem B/Theorem 3.4. in "Exactness of locally compact groups" http://arxiv.org/abs/1603.01829