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A gap theorem for the ZL-amenability constant of a finite group. (English) Zbl 1454.43002
Summary: It was shown in [A. Azimifard et al., J. Funct. Anal. 256, No. 5, 1544–1564 (2009; Zbl 1167.43004)] that the ZL-amenability constant of a finite group is always at least 1, 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 301/300, but the proof relies crucially on a deep result of D. Rider [Trans. Am. Math. Soc. 186, 459–479 (1974; Zbl 0274.43002)] 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 7/4, which is known to be best possible. We avoid use of Rider’s result [loc. cit.], by analyzing the cases where $$G$$ is just non-abelian, using calculations from [M. Alaghmandan et al., Can. Math. Bull. 57, No. 3, 449–462 (2014; Zbl 1300.43004)], and establishing a new estimate for groups with trivial centre.
##### MSC:
 43A20 $$L^1$$-algebras on groups, semigroups, etc. 20C15 Ordinary representations and characters
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##### References:
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