Weak amenability of weighted group algebras

In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G, \omega)$ is weakly amenable if and only if every non-inner quasi-additive function in $L^\infty(G, 1/\omega)$ is unbounded. This provides an answer to the question concerning weak amenability of $L^1(G, \omega)$ and improve some known results in connection with it

Weak amenability of weighted measure algebras and their second duals

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 $M(G, \omega)^{**}$