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

It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 256 (2009)] that the ZL amenability constant of a finite group is always at least~1 1 , with equality if and only if the group is abelian. It was also shown in [A. Azimifard, E. Samei, N. Spronk, op cit.] that for any finite non-abelian group this invariant is at least 301/300 301/300 , 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 G is finite and non-abelian then its ZL-amenability constant is at least 7/4 7/4 , which is known to be best possible. We avoid use of Rider's reslt, by analyzing the cases where G G is just non-abelian, using calculations from [M. Alaghmandan, Y. Choi, E. Samei, CMB 57 2014)], and establishing a new estimate for groups with trivial centre.