$(\sigma, \sigma)$-derivation and $(\sigma, \tau)$-weak amenability of Beurling algebra

Bull. Korean Math. Soc. Published online February 24, 2021

Lin Chen and Jianhua Zhang
Shannxi Normal University

Abstract : Let $G$ be a topological group with a locally compact and Hausdorff
topology. $\omega$ is a diagonally
bounded weight. In this paper, the generalized notions of derivation and weak amenability, namely $(\sigma,\tau)$-derivation and $(\sigma,\tau)$-weak amenability,
where $\sigma,\tau$ are isometric automorphism of Banach algebra, are studied on Beurling algebra $L^1_{\omega}(G)$.
We prove that every continuous $(\sigma,\sigma)$-derivation from $L^1_{\omega}(G)$ into measure algebra $M_{\omega}(G)$
is $(\sigma,\sigma)$-inner and Beurling algebra $L^1_{\omega}(G)$ is $(\sigma,\tau)$-weakly amenable.