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Uniformly quasi-Hermitian groups are supramenable

Part of: Abstract harmonic analysis Selfadjoint operator algebras

Published online by Cambridge University Press:  14 July 2021

Mahmud Azam  and
Ebrahim Samei
Mahmud Azam
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SKS7N 5E6, Canada e-mail: [email protected]
Ebrahim Samei*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SKS7N 5E6, Canada e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Motivated by the recent result in Samei and Wiersma (2020, Advances in Mathematics 359, 106897) that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

Keywords

Qausi-Hermitian groupsamenable groupssupramenable groupsgroups with subexponential growthuniform Roe algebras

MSC classification

Primary: 43A20: $L^1$-algebras on groups, semigroups, etc.
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 46L55: Noncommutative dynamical systems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 665 - 673
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first-named author was partially supported by NSERC USRA 2020. The second-named author was partially support by NSERC Grant no. 409364-2019.

References

Barnes, A. B., When is the spectrum of a convolution operator on Lp independent of p? Proc. Edinb. Math. Soc. (2) 33(1990), no. 2, 327–332.CrossRefGoogle Scholar
Brodzki, J., Niblo, G. A., and Wright, N. J., Property A, partial translation structures, and uniform embeddings in groups. J. Lond. Math. Soc. (2) 76(2007), no. 2, 479–497.CrossRefGoogle Scholar
Ceccherini-Silberstein, T., Grigorchuk, R., and de la Harpe, P., Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Proc. Steklov Inst. Math. 224(1999), no. 1, 57–97.Google Scholar
Chung, Y. C. and Li, K., Ridigity of ${\ell}^{\mathrm{p}}$ Roe-type algebras. Bull. Lond. Math. Soc. 50(2018), no. 6, 1056–1070.CrossRefGoogle Scholar
Fendler, G., Gröchenig, K., and Leinert, M., Symmetry of weighted ${\mathrm{L}}^1$ -algebras and the GRS-condition. Bull. Lond. Math. Soc. 38(2006), no. 4, 625–635.CrossRefGoogle Scholar
Hulanicki, A., On the spectrum of convolution operators on groups with polynomial growth. Invent. Math. 17(1972), 135–142.CrossRefGoogle Scholar
Jenkins, J. W., Symmetry and nonsymmetry in the group algebras of discrete groups. Pacific J. Math. 32(1970), 131–145.CrossRefGoogle Scholar
Kellerhals, J., Monod, N., and Rørdam, M., Non-supramenable groups acting on locally compact spaces. Doc. Math. 18(2013), 1597–1626.Google Scholar
Losert, V., On the structure of groups with polynomial growth II. J. Lond. Math. Soc. 63(2001), 640–654.CrossRefGoogle Scholar
Palma, R., Quasi-symmetric group algebras and C*-completions of Hecke algebras. In: Operator algebra and dynamics, Springer Proc. Math. Stat., 58, Springer, Heidelberg, 2013, pp. 253–271.CrossRefGoogle Scholar
Palmer, T. W., Banach algebras and the general theory of $\ast$ -algebras. Vol. 2. *-algebras, Encycl. Math. Appl., 79, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
Samei, E. and Wiersma, M., Quasi-Hermitian locally compact groups are amenable. Adv. Math. 359(2020), 106897, 25 pages.CrossRefGoogle Scholar