An example of a weighted algebra $L_p^w(G)$ on uncountable group

TL;DR

This paper constructs weighted algebra examples on uncountable free groups for certain p-values, showing their non-existence for p>2 and linking their existence to sigma-compactness of amenable groups.

Contribution

It provides the first examples of weighted algebras on uncountable free groups for 1<p≤2 and establishes conditions for their existence on amenable groups.

Findings

Weighted algebras exist on uncountable free groups for 1<p≤2.

No weighted algebras exist on these groups for p>2.

Existence of weighted algebras with p>1 implies the group is sigma-compact.

Abstract

We construct examples of weighted algebras $L_p^w(G)$ with $1<p\le 2$ on uncountable free groups. For $p>2$ no weighted algebras exist on these groups. From the other side, we prove that an amenable group on which exist weighted algebras with $p>1$ must be sigma-compact.