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Weight functions on groups and an amenability criterion for Beurling algebras. (English. Russian original) Zbl 0905.43001
Math. Notes 60, No. 3, 274-282 (1996); translation from Mat. Zametki 60, No. 3, 370-382 (1996).
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.

43A20 \(L^1\)-algebras on groups, semigroups, etc.
20J05 Homological methods in group theory
20D15 Finite nilpotent groups, \(p\)-groups
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