Geodesic
Weight functions on groups and an amenability criterion for Beurling algebras
Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 370-382.

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The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebra $l^1(G,\omega)$ constructed for the weight $\omega$ is amenable if and only if the group $G$ is amenable and the weight $\omega$ is equivalent to a multiplicative character $\chi\colon G\to\mathbb R_+$. 
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R. I. Grigorchuk. Weight functions on groups and an amenability criterion for Beurling algebras. Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 370-382. http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a4/

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