Examples of groups which are not weakly amenable (Q428131)

Let \(G\) be a locally compact group. It is \textit{weakly amenable} if it has a completely bounded approximate identity. The author proves that if \(G\) is weakly amenable then there exists a \((G\ltimes N)\)-invariant state on \(L^\infty(N)\) for any amenable closed normal subgroup \(N\subset G\). This provides new examples of non-amenable groups: let \(\Gamma\) and \(\Lambda\) be discrete groups, \(\Lambda\) nontrivial and \(\Gamma\) nonamenable, then their wreath product \(\Lambda\wr\Gamma\) is not weakly amenable. A von Neuman algebra analogue is also obtained.