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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.

##### MSC:
 43A20 $$L^1$$-algebras on groups, semigroups, etc. 20J05 Homological methods in group theory 20D15 Finite nilpotent groups, $$p$$-groups
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