Some notions of $(sigma, tau)-$amenability for unitization of Banach algebras

Abstract

Let $mathcal A$ be a Banach algebra and $sigma$ and $tau$ be continuous endomorphisms on $mathcal A$. In this paper, we investigate $(sigma, tau)-$amenability and<br />$(sigma, tau)-$weak amenability for unitization of Banach algebras, and also the relation between of them.<br />We introduce and study the concepts $(sigma, tau)-$trace extention property, $(sigma, tau)-I-$weak amenability and $(sigma, tau)-$ideal amenability for $mathcal A$ and its unitization, where $I$ is a closed two-sided ideal in $mathcal A$.