Bulletin of the
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Bull. Korean Math. Soc. 2021; 58(5): 1209-1219

Online first article February 24, 2021      Printed September 30, 2021

https://doi.org/10.4134/BKMS.b200873

Copyright © The Korean Mathematical Society.

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

Lin Chen, Jianhua Zhang

Anshun University; Shaanxi Normal University

Abstract

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

Keywords: $(\sigma,\sigma)$-derivation, $(\sigma,\tau)$-weak amenability, Beurling algebras

MSC numbers: 47B49, 46K15

Supported by: This work is supported by the National Natural Science Foundation of China (No. 12061018) and the Postdoctoral Science Foundation of China (No. 2018M633450). The first author is supported by Foundation of Educational Commission (No. KY[2017]092) and of Science and Technology department (No. [2018]1001) of Guizhou Province of China.

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