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A gap theorem for the ZL-amenability constant of a finite group | ||
| International Journal of Group Theory | ||
| مقاله 34، دوره 5، شماره 4، اسفند 2016، صفحه 27-46 اصل مقاله (261.98 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.9562 | ||
| نویسنده | ||
| Yemon Choi* | ||
| Lancaster University | ||
| چکیده | ||
| It was shown in [A. Azimifard, E. Samei and N. Spronk, Amenability properties of the centres of group algebras, J. Funct. Anal., 256 no. 5 (2009) 1544-1564.] 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. 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 $7/4$, 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., 57 (2014) 449-462.], and establishing a new estimate for groups with trivial centre. | ||
| کلیدواژهها | ||
| Amenability constant؛ character degrees؛ just non-abelian groups | ||
| مراجع | ||
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[1] M. Alaghmandan, Y. Choi and E. Samei, ZL-amenability constants of nite groups with two character degrees, Canad. Math. Bul l., 57 (2014) 449-462. [2] A. Azimifard, E. Samei and N. Spronk, Amenability prop erties of the centres of group algebras, J. Funct. Anal., 256 (2009) 1544-1564. [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 nite groups, Corrected reprint of the 1976 original, Academic Press, New York, Dover Publications, Inc., New York, 1994. [7] G. James and M. Lieb eck, Representations and characters of groups, second ed., Cambridge University Press, New York, 2001. [8] M. F. Newman, On a class of metab elian groups, Proc. London Math. Soc. (3), 10 (1960) 354-364. [9] M. F. Newman, On a class of nilp otent groups, Proc. London Math. Soc. (3), 10 (1960) 365-375. [10] D. Rider, Central idemp otent measures on compact groups, Trans. Amer. Math. Soc., 186 (1973) 459-479. | ||
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