A gap theorem for the ZL-amenability constant of a finite group

It was shown in [‎A‎. ‎Azimifard‎, ‎E‎. ‎Samei and N‎. ‎Spronk‎, ‎Amenability properties of the centres of group algebras‎, J‎. ‎Funct‎. ‎Anal.‎, ‎۲۵۶ no‎. ‎۵ (۲۰۰۹) ۱۵۴۴-۱۵۶۴‎.] that the ZL-amenability constant of a finite group is always at least ۱‎, ‎with equality if and only if the group is abelian‎. ‎It was also shown that for any finite non-abelian group this invariant is at least ۳۰۱/۳۰۰‎, ‎but the proof relies crucially on a deep result of D‎. ‎A‎. ‎Rider on norms of central idempotents in group algebras‎.  ‎Here we show that if G is finite and non-abelian then its ZL-amenability constant is at least ۷/۴‎, ‎which is known to be best possible‎. ‎We avoid use of Rider's reslt‎, ‎by analyzing the cases where G is just non-abelian‎, ‎using calculations from [‎M‎. ‎Alaghmandan‎, ‎Y‎. ‎Choi and E‎. ‎Samei‎, ‎ZL-amenability constants of finite groups with two character degrees‎, Canad‎. ‎Math‎. ‎Bull.‎, ‎۵۷ (۲۰۱۴) ۴۴۹-۴۶۲‎.]‎, ‎and establishing a new estimate for groups with trivial centre‎.