Weight functions on groups and an amenability criterion for Beurling algebras

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 $\chi\colon G\to\mathbb R_+$.