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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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[1] M. Alaghmandan, Y. Choi and E. Samei, ZL-amenability constants of finite groups with two character degrees, Canad. Math. Bull.,57(2014) 449-462. · Zbl 1300.43004
[2] A. Azimifard, E. Samei and N. Spronk, Amenability properties of the centres of group algebras,J. Funct. Anal., 256(2009) 1544-1564. · Zbl 1167.43004
[3] P. M. Cohn,Algebra.,1, second ed., John Wiley & Sons Ltd., Chichester, 1982.
[4] P. M. Cohn,Algebra.,2, second ed., John Wiley & Sons Ltd., Chichester, 1989.
[5] P. M. Cohn,Algebra.,3, second ed., John Wiley & Sons Ltd., Chichester, 1991.
[6] I. M. Isaacs,Character theory of finite groups, Corrected reprint of the 1976 original, Academic Press, New York, Dover Publications, Inc., New York, 1994.
[7] G. James and M. Liebeck,Representations and characters of groups, second ed., Cambridge University Press, New York, 2001. · Zbl 0981.20004
[8] M. F. Newman, On a class of metabelian groups,Proc. London Math. Soc. (3),10(1960) 354-364. · Zbl 0099.25202
[9] M. F. Newman, On a class of nilpotent groups,Proc. London Math. Soc. (3),10(1960) 365-375. · Zbl 0099.25203
[10] D. Rider, Central idempotent measures on compact groups,Trans. Amer. Math. Soc.,186(1973) 459-479. Yemon Choi Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom Email:y.choi.97@cantab.net · Zbl 0274.43002
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