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 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.