Example of a weighted algebra on an uncountable discrete group

We construct examples of weighted algebras L p w ( G ) with 1 < p ⩽ 2 on uncountable free groups. For p > 2 no weighted algebras exist on these groups. On the other hand, we prove that for any amenable locally compact group G, if L p w ( G ) is an algebra for some weight w and p > 1 , then G is σ-compact.