Weight functions on groups and an amenability criterion for Beurling algebras (Q1387332)

The paper studies semiweights on (discrete) groups, i.e. functions which, for elements \(g,h\), satisfy \(\omega (gh)\leq c\omega (g) \omega(h)\), with some absolute constant \(c>0\). It is shown in Theorem 1 that (weak equivalence classes of) logarithms of semiweights form a real vector space isomorphic to \(H^*_{b,2} (G)\), the singular part of the bounded cohomology group \(H^*_b (G)\). The author also proves in Theorem 2 that a Beurling (Banach convolutive) algebra \(l^1 (G,\omega)\) is amenable iff the group is amenable and the weight \(\omega\) is equivalent to a positive character on the group \(G\). Moreover, the relations of weights to the Tychonoff property of groups and to harmonic functions on groups are investigated.