Weak amenability of weighted measure algebras and their second duals

TL;DR

This paper investigates the conditions under which weighted measure algebras and their second duals are weakly amenable, linking this property to the discreteness and finiteness of the underlying group.

Contribution

It establishes necessary and sufficient conditions for weak amenability of weighted measure algebras and their second duals, providing answers to open questions in the field.

Findings

Weak amenability of $M(G, ext{ω})$ iff $G$ is discrete and all bounded quasi-additive functions are inner.

Weak amenability of $L^1(G, ext{ω})^{**}$ and $M(G, ext{ω})^{**}$ iff $G$ is finite.

The results clarify the relationship between group properties and the weak amenability of associated Banach algebras.

Abstract

In this paper, we study the weak amenability of weighted measure algebras and prove that $M(G, \omega)$ is weakly amenable if and only if $G$ is discrete and every bounded quasi-additive function is inner. We also study the weak amenability of $L^1(G, \omega)^{**}$ and $M(G, \omega)^{**}$ and show that the weak amenability of theses Banach algebras are equivalent to finiteness of $G$. This gives an answer to the question concerning weak amenability of $L^1(G, \omega)^{**}$ and $M(G, \omega)^{**}$.