Semi-amenability of Banach algebras

Mostfa Shams Kojanaghi, Kazem Haghnejad Azar

Published 2019-05-26Version 1

Let $\mathcal{A}$ be a Banach algebra, and $\mathcal{X}$ be a Banach $\mathcal{A}$-bimodule. A derivation $\mathcal{D}:\mathcal{A}\rightarrow \mathcal{X}$ from Banach algebra into Banach space is called semi-inner if there are $\eta , \xi \in \mathcal{X}$ such that $$ \mathcal{D}(a)=a.\eta-\xi.a=\delta_{\eta,\xi}(a), \;\;\;\;\; (a\in \mathcal{A}).$$ A Banach algebra $\mathcal{A}$ is semi-amenable (resp. semi-contractible) if, for each Banach $\mathcal{A}$-bimodule $\mathcal{X}$, every derivation $\mathcal{D}$ from $\mathcal{A}$ into $\mathcal{X}^{*}$ (resp. into $\mathcal{X}$) is semi-inner. In this paper, we study some problems in semi-amenability of Banach algebras which have been studied in amenability case. We extend some definitions and concepts for semi-amenability, that is, we introduce approximately semi-amenability, semi-contractibility with solving some problems which former have been studied for amenability case.